Math 2.0

statement proposition

Sum of a finite geometric series

Statement

Let rr be a real number with r≠1r \ne 1, and let nn be a natural number. Then ∑k=0n−1rk=1+r+r2+⋯+rn−1=1−rn1−r.\sum_{k=0}^{n-1} r^k = 1 + r + r^2 + \cdots + r^{n-1} = \frac{1 - r^n}{1 - r}.

Remarks

The proof multiplies the sum SS by 1−r1 - r. The product (1−r)S(1 - r) S telescopes to 1−rn1 - r^n. For r=1r = 1 the sum is nn. The identity holds in each commutative ring in the form (1−r)∑k=0n−1rk=1−rn(1 - r) \sum_{k=0}^{n-1} r^k = 1 - r^n, with no division. Mathlib also states the version for a sum over the indices m≤k<nm \le k < n, as geom_sum_Ico. Letting n→∞n \to \infty gives the geometric series.

Explanations

L1 human

Write S=1+r+⋯+rn−1S = 1 + r + \cdots + r^{n-1} and rS=r+r2+⋯+rnrS = r + r^2 + \cdots + r^n. Subtract, and almost all terms cancel: S−rS=1−rnS - rS = 1 - r^n. For example, 1+2+4+8=(1−16)/(1−2)=151 + 2 + 4 + 8 = (1 - 16)/(1 - 2) = 15.

What this node needs for W1
  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 4 nodes in this order. The foundations come first, and the list ends with this node.

Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionSum of a finite geometric series, statement, no evidence levelSum of a finite geome…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Sequence definition
  2. Real sequence definition
  3. Geometric sequence definition
  4. Sum of a finite geometric series statement

What depends on this

83 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Geometric_series, Definition: finite geometric series

What links here

Used by