Does the infinite series Σ_(n=1)^(∞) ((-1)^n+1)/n converge? If yes, find its sum.

Does the infinite series $n = 1 \sum \infty \frac{( - 1 ) ^{n + 1}}{n}$ converge? If yes, find its sum.

hard 4 marks Class 11 · Mathematics · Sequences and Series

Answer

ln 2

The correct answer is ln 2 — Hence the sum of the series is $l n 2$ .

Identify the series as the alternating harmonic series: terms are $\frac{( - 1 ) ^{n + 1}}{n}$ .
Apply the Alternating Series Test (Leibniz criterion). The sequence $b_{n} = \frac{1}{n}$ is positive, decreasing, and $lim_{n \to \infty} b_{n} = 0$ .
Therefore the series converges.
To find its sum, recognize the Taylor series of $l n (1 + x)$ at $x = 1$ : $l n (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots$ for $∣ x ∣ l e 1, x e q - 1$ .
Substituting $x = 1$ gives $l n 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ , which is exactly our series.
Hence the sum of the series is $l n 2$ .

Common mistake: Using the ratio test and concluding divergence or forgetting that the alternating harmonic converges to

l n 2

Concept tested: Alternating series convergence