Question

Consider the series $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$. Let $S_k := \sum_{n=1}^{k} \frac{1}{n(n+1)}$ be the kth partial sum. (a) (12 points) Prove $S_k = 1 - \frac{1}{k+1}$ using a proof by induction.

          Consider the series $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$. Let $S_k := \sum_{n=1}^{k} \frac{1}{n(n+1)}$ be the kth partial sum.
(a) (12 points) Prove $S_k = 1 - \frac{1}{k+1}$ using a proof by induction.

Consider the series ∑n=1^∞(1)/(n(n+1)). Let Sk := ∑n=1^k(1)/(n(n+1)) be the kth partial sum.
(a) (12 points) Prove Sk = 1 - (1)/(k+1) using a proof by induction.

Added by Lance L.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Nick Johnson

11/09/2021

Step 1

In this case, the partial sum S is given by S = Σ(n/(n+1)) from n = 1 to 1. Plugging in n = 1, we get S = 1/(1+1) = 1/2. Show more…

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Text: Can someone please help? I'll like it if corrected. Preferably, writing down the answer helps! Thank you. 8.1 Consider the series. Let S be the kth partial sum, given by S = Σ(n/(n+1)), where n ranges from 1 to k. Prove that S = 1 using a proof by induction.