a good infinity

Calculus Level 2

1 − 1 2 + 1 3 − 1 4 + 1 5 − 1 6 + … 1 - \frac{1}{2} +\frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \ldots

0 ln ⁡ 2 \ln 2 ln ⁡ π \ln \pi ln ⁡ 1 \ln 1

Kaivalya Swami by kaivalya swami , 21, India

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1 solution

Andrew Machkasov
Apr 26, 2015

We know that the power series for 1/(1+x) is 1 + x + x^2 + x^3 + ... if |x| < 1, and thus the series x + (x^2)/2 + (x^3)/3 + ... is the series for ln(1+x) for |x|< 1. However, the series also converges at x = 1 due to the fact that it's alternating and strictly decreasing. Thus the series is ln(1 + 1) = ln(2).

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