Proof of the Leibniz formula for pi by integrating a finite geometric sum

Builds on Alternating series test pi Arctangent Tangent function Sine and cosine Angle addition formulas Intermediate value theorem Derivative of sine Mean value theorem Derivative of arctangent Differentiable functions are continuous Continuous functions are Riemann integrable Second fundamental theorem of calculus Sum of a finite geometric series Linearity of the integral Integral of a power Monotonicity of the integral Limit of a sequence

Proof

Let $s_N = \sum_{n=0}^{N} \frac{(-1)^n}{2n+1}$ be the partial sums.

The numbers $b_n = 1/(2n+1)$ are positive, decreasing, and tend to $0$ . By the alternating series test, the series converges.
We compute $\arctan 1$ . By the definition of pi, $\pi/2$ is the only zero of $\cos$ in $[0, 2]$ . Also $\pi/2 \ne 0$ , since $\cos 0 = 1$ by the definition of sine and cosine.
The function $\cos$ is continuous and $\cos 0 = 1 > 0$ . If $\cos x \le 0$ for some $x \in [0, \pi/2)$ , the intermediate value theorem would give a zero of $\cos$ in $(0, x]$ . This zero would lie in $[0, 2]$ and differ from $\pi/2$ , which is impossible. So $\cos x > 0$ on $[0, \pi/2)$ .
By the mean value theorem and the derivative of sine, $\sin(\pi/4) - \sin 0 = \frac{\pi}{4} \cos \xi$ for some $\xi \in (0, \pi/4)$ . By step 3 this is positive, and $\sin 0 = 0$ , so $\sin(\pi/4) > 0$ .
By the angle addition formulas, $0 = \cos(\pi/2) = \cos^2(\pi/4) - \sin^2(\pi/4)$ . Since $\sin(\pi/4)$ and $\cos(\pi/4)$ are both positive, they are equal.
So $\tan(\pi/4) = 1$ , by the definition of the tangent function. Since $\pi/4 \in (-\pi/2, \pi/2)$ , the definition of the arctangent gives $\arctan 1 = \pi/4$ .
Also $\tan 0 = \sin 0 / \cos 0 = 0$ , so $\arctan 0 = 0$ .
The arctangent is differentiable with derivative $1/(1+x^2)$ , by the derivative of arctangent. So it is continuous. Its derivative is a quotient of continuous functions with $1 + x^2 > 0$ , so it is continuous, hence Riemann integrable on $[0, 1]$ .
By the second fundamental theorem of calculus and steps 6 and 7, $\int_0^1 \frac{dt}{1 + t^2} = \arctan 1 - \arctan 0 = \frac{\pi}{4}.$
Let $n \ge 1$ and $t \in [0, 1]$ . Apply the sum of a finite geometric series with $r = -t^2 \ne 1$ . It gives $\sum_{k=0}^{n-1} (-1)^k t^{2k} = \frac{1 - (-1)^n t^{2n}}{1 + t^2}$ .
Rearranging step 10 gives $\frac{1}{1 + t^2} = \sum_{k=0}^{n-1} (-1)^k t^{2k} + (-1)^n \frac{t^{2n}}{1 + t^2}.$
All functions in step 11 are continuous on $[0,1]$ , hence integrable. By linearity of the integral and the integral of a power, $\int_0^1 t^{2k}\, dt = \frac{1}{2k+1}$ . Integrating step 11 over $[0, 1]$ and using step 9 gives $\frac{\pi}{4} = s_{n-1} + (-1)^n R_n, \qquad R_n = \int_0^1 \frac{t^{2n}}{1 + t^2}\, dt.$
For $t \in [0, 1]$ , $0 \le \frac{t^{2n}}{1 + t^2} \le t^{2n}$ . By monotonicity of the integral and the integral of a power, $0 \le R_n \le \frac{1}{2n+1}$ .
By steps 12 and 13, $\abs{\pi/4 - s_{n-1}} = R_n \le \frac{1}{2n+1}$ for every $n \ge 1$ .
Let $\varepsilon > 0$ , and take $n_0$ with $1/(2n_0 + 1) < \varepsilon$ . For $N \ge n_0 - 1$ , step 14 with $n = N + 1$ gives $\abs{\pi/4 - s_N} \le 1/(2N+3) < \varepsilon$ .
So $s_N \to \pi/4$ by the definition of the limit. Hence $1 - \frac13 + \frac15 - \frac17 + \cdots = \frac{\pi}{4}$ .

Idea

Since $\arctan' = 1/(1+t^2)$ , the number $\pi/4 = \arctan 1$ is the integral of $1/(1+t^2)$ over $[0,1]$ . Expanding $1/(1+t^2)$ as a finite geometric sum in $-t^2$ plus a remainder, and integrating term by term, gives a partial sum of the series. The remainder integral is at most $1/(2n+1)$ , so it tends to $0$ .

Reading path

To understand this page from the ground up, read these 299 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the Cauchy–Hadamard theorem, Proof of the derivative of cosine by term-by-term differentiation, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the formula for the derivative of a polynomial, Proof of the reciprocal rule, Proof of the commutativity of addition by induction, Proof of term-by-term differentiation, Proof of the integral of a power, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof of the reverse triangle inequality, Proof of the limit of a quotient of sequences, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the alternating series test, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof of the derivative of arctangent, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of Pascal's rule from the recursive definition, Proof of the linearity of differentiation, Proof of the angle addition formulas by differentiation, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the derivative of tangent by the quotient rule, Proof of the epsilon-N definition of the limit of a sequence, Proof of the quotient rule from the product and reciprocal rules, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the continuity of the inverse of a strictly monotone continuous function, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the linearity of the integral, Proof of the factorial formula for binomial coefficients by induction, Proof that a function with positive derivative is strictly increasing, Proof of the second fundamental theorem of calculus, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof that one-sided limits of a monotone function exist, Proof of the epsilon-delta form of the limit of a function, Proof of the derivative of sine by term-by-term differentiation, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a composition of continuous functions is continuous, Proof of the Pythagorean trigonometric identity by differentiation, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that a function with zero derivative on an open interval is constant, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

Binomial coefficient definition
Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Factorial definition
Group concept
Integers object
Limit point of a set definition
Local maximum definition
Mathematical induction concept
Natural numbers object
Ordered field definition
Partition of an interval definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Strictly monotone function definition
Proof of Pascal's rule from the recursive definition proof
Set definition
Ring definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Archimedean ordered field definition
Successor function definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Pascal's rule statement
Ordered pair definition
Commutative ring definition
Continuity at a point definition
Derivative definition
One-sided limit definition
Addition of natural numbers definition
Proof of the Archimedean property in an Archimedean ordered field proof
Riemann integrable function definition
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Truth table concept
Modus ponens concept
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Sequence tending to infinity definition
Series definition
Completeness axiom statement
Cartesian product definition
Proof of the factorial formula for binomial coefficients by induction proof
Continuous function definition
Differentiable function definition
One-sided derivative definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
Riemann integral definition
Subset definition
The rationals form an ordered field statement
Propositional logic concept
Mathematical proof concept
Upper and lower Darboux sums definition
Uniqueness quantifier definition
Limit superior definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Pointwise convergence definition
Sequential continuity definition
The sequence r^n tends to infinity for r > 1 example
Alternating series definition
Cauchy product definition
Partial sum definition
Power series definition
Factorial formula for binomial coefficients statement
Proof that a composition of continuous functions is continuous proof
Binary relation definition
Order on the reals definition
Commutativity of addition statement
Definite integral definition
Proof of strong induction from ordinary induction proof
Tautology definition
Upper and lower Darboux integrals definition
Sum of a finite geometric series statement
Uniform convergence definition
Radius of convergence definition
Convergent series definition
Sine and cosine definition
Proof of the binomial theorem by induction proof
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Interval definition
Strong induction statement
Function definition
Uniform convergence on a set definition
Logical equivalence definition
Tangent function definition
Series of functions definition
Absolute convergence definition
Sum of a series definition
Binomial theorem statement
Preorder definition
Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
Codomain definition
Injective function definition
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Epsilon-delta definition of continuity statement
Epsilon-delta definition of a limit statement
Monotone function definition
Surjective function definition
Well-ordering principle statement
Double negation law statement
Total order definition
Uniform continuity definition
Composition of continuous functions is continuous statement
Bijective function definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Uniform continuity on a set definition
Inverse function definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Existence of the infimum statement
Triangle inequality statement
Absolute value is multiplicative statement
Epsilon-N definition of the limit of a sequence statement
Proof of the intermediate value theorem by the supremum proof
Proof that one-sided limits of a monotone function exist proof
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the reverse triangle inequality proof
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
Proof that differentiable functions are continuous proof
Intermediate value theorem statement
Proof of the linearity of the integral proof
The reals are Archimedean statement
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
Proof that subsequences of a convergent sequence converge to the same limit proof
The sequence 1/n tends to zero example
Proof that Cauchy sequences are bounded proof
Proof of the term test proof
Proof of the monotone convergence theorem for sequences proof
Reverse triangle inequality statement
Differentiable functions are continuous statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Proof of the continuity of the inverse of a strictly monotone continuous function proof
Linearity of the integral statement
Limit of a sum of sequences statement
Uniqueness of limits statement
Proof of the monotonicity of the integral proof
Limits preserve non-strict inequalities statement
Squeeze theorem statement
Proof that Darboux and Riemann integrals agree proof
Convergent sequences are bounded statement
Subsequences of a convergent sequence converge to the same limit statement
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Proof of the limit of a product of sequences proof
There is a natural number n with 1/n below any positive epsilon statement
Monotonicity of the integral statement
One-sided limits of a monotone function exist statement
Darboux and Riemann integrals agree statement
Term test statement
Proof of the alternating series test proof
Proof that a decreasing sequence bounded below converges proof
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
The sequence r^n tends to zero for |r| < 1 example
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Continuity of the inverse of a strictly monotone continuous function statement
Limit of a product of sequences statement
Decreasing sequence bounded below converges statement
Proof of Riemann's criterion for integrability proof
Geometric series example
A series of nonnegative terms converges iff its partial sums are bounded statement
Alternating series test statement
Proof of the limit of a quotient of sequences proof
Arctangent definition
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Sequential characterization of limits statement
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Proof of the root test proof
Proof of the Cauchy–Hadamard theorem proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the ratio test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the comparison test proof
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Limit of a quotient of sequences statement
Comparison test statement
Cauchy product of absolutely convergent series statement
Bolzano–Weierstrass theorem statement
Exponentials dominate powers in sequences statement
Chain rule statement
Product rule statement
Proof of the reciprocal rule proof
Linearity of differentiation statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Ratio test statement
Root test statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Derivative of an inverse function statement
Reciprocal rule statement
Proof of the power rule by induction proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof of the constant multiple rule proof
Boundedness theorem statement
Cauchy–Hadamard theorem statement
Cauchy criterion for sequences statement
Power rule statement
Proof of the quotient rule from the product and reciprocal rules proof
Constant multiple rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Quotient rule statement
Proof of the formula for the derivative of a polynomial proof
Extreme value theorem statement
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof that continuous functions are Riemann integrable proof
Fermat's theorem on stationary points statement
Proof that absolutely convergent series converge proof
Continuous functions are Riemann integrable statement
Proof of Rolle's theorem proof
Absolutely convergent series converge statement
Rolle's theorem statement
Proof of the Weierstrass M-test proof
Proof of the mean value theorem from Rolle's theorem proof
Weierstrass M-test statement
Mean value theorem statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Proof that a function with zero derivative on an open interval is constant proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Proof that a function with positive derivative is strictly increasing proof
Proof of the second fundamental theorem of calculus proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
A function with zero derivative is constant statement
Uniform convergence of derivatives allows term-by-term differentiation statement
Second fundamental theorem of calculus statement
A function with positive derivative is strictly increasing statement
Proof of the integral of a power proof
Proof of term-by-term differentiation proof
Integral of a power statement
Term-by-term differentiation statement
Proof that power series are differentiable inside the radius of convergence proof
Power series are differentiable inside the radius of convergence statement
Proof of the derivative of cosine by term-by-term differentiation proof
Proof of the derivative of sine by term-by-term differentiation proof
Derivative of cosine statement
Derivative of sine statement
Proof of the angle addition formulas by differentiation proof
Proof of the Pythagorean trigonometric identity by differentiation proof
pi object
Pythagorean trigonometric identity statement
Angle addition formulas statement
Proof of the derivative of tangent by the quotient rule proof
Derivative of tangent statement
Proof of the derivative of arctangent proof
Derivative of arctangent statement
Proof of the Leibniz formula for pi by integrating a finite geometric sum proof

Where this leads

2 pages build on this one.

Tag	`2703AZH0B6`
URI	`m2://2703AZH0B6`
Kind	proof
Writing	W0 stub
MSC	40A05
Proves	Leibniz formula for pi
Uses	Alternating series test, pi, Arctangent, Tangent function, Sine and cosine, Angle addition formulas, Intermediate value theorem, Derivative of sine, Mean value theorem, Derivative of arctangent, Differentiable functions are continuous, Continuous functions are Riemann integrable, Second fundamental theorem of calculus, Sum of a finite geometric series, Linearity of the integral, Integral of a power, Monotonicity of the integral, Limit of a sequence
Changed	4 November 2026

Proof of the Leibniz formula for pi by integrating a finite geometric sum

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Idea

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