Mathematics

Problem

Consider the infinite series

S=\sum_{n=1}^{\infty}(-1)^{n+1}\,\frac{1}{n+1}.

Let

S_5

denote the partial sum obtained by adding the first five terms of the series. a. Verify that the sequence of term magnitudes

\left\{\dfrac{1}{n+1}\right\}_{n\ge 1}

is positive, eventually decreasing, and converges to

0

. Provide clear reasoning for each of the three properties. b. Compute the magnitude of the next term that would be added after

S_5

, namely

|a_6|

. c. Explain what your result from part (b) implies about the largest possible difference between

S

and

S_5

. State your conclusion in one complete sentence that links the calculation to the accuracy of

S_5

as an approximation of

S

Intuitive explanation

This series alternates in sign, so its partial sums oscillate around the true value—understanding this behavior is key. The trick is recognizing that $S_5$ gives you a specific rational number by carefully tracking which terms are positive and which are negative. Once you compute $S_5$ , you can compare it to the infinite sum by noticing a connection to the logarithm series $\ln(1+x)$ , which will help you see the big picture of where this series converges.

Intuitive explanation

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