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  "body": "\n## Example\n\nLet $a$ and $r$ be real numbers with $|r| \u003c 1$.\nThe geometric series $\\sum_{n=0}^{\\infty} a r^n$ converges, and\n$$\\sum_{n=0}^{\\infty} a r^n = \\frac{a}{1 - r}.$$\nThe [[sum of a finite geometric series|05APWSX53S]] gives the partial sums $s_N = a (1 - r^{N+1})/(1 - r)$.\nSince [[$r^{N+1} \\to 0$|0Q7K4AXE53]], the partial sums tend to $a/(1 - r)$.\n\n## Remarks\n\nFor $a \\ne 0$ and $|r| \\ge 1$, the series diverges by the [[term test|0VVY4H92G2]], because $|a r^n| \\ge |a|$.\nFor $a = 1$ and $r = 1/2$, it gives $1 + 1/2 + 1/4 + \\cdots = 2$.\nThe decimal $0.333\\ldots$ is the geometric series with $a = 3/10$ and $r = 1/10$, and its sum is $1/3$.\nThe terms form a [[geometric sequence|0MNM54WB99]].\n\n## Explanations\n\n### L1 human\n\nWalk half of the way to a wall, then half of the rest, and so on.\nThe distances $1/2 + 1/4 + 1/8 + \\cdots$ add up to the whole distance $1$.\n",
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