Find Sn for the serks到。Does the series converge

Consider the given series. Sn=∑n=1∞1n+3-1n+4 Expend the series and find the limit…

Answered: Find Sn for the serks n= Does the seres…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter8: Sequences, Series, And Probability

Section8.4: Geometric Sequences And Series

Problem 27E

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Find Sn for the serks
到。
Does the series converge

Expert Solution

Step 1

Consider the given series.

$S_{n} = \sum_{n = 1}^{\infty} (\frac{1}{\sqrt{n + 3}} - \frac{1}{\sqrt{n + 4}})$

Expend the series and find the limit

$\begin{array}{rcl} _{n = 1}^{\infty} (\frac{1}{\sqrt{n + 3}} - \frac{1}{\sqrt{n + 4}}) & = & \lim_{n \to \infty} [\frac{1}{\sqrt{4}} - \frac{1}{\sqrt{5}} + \frac{1}{\sqrt{5}} - \frac{1}{\sqrt{6}} + \frac{1}{\sqrt{6}} + \dots - \frac{1}{\sqrt{n + 1}} + \frac{1}{\sqrt{n + 1}} - \frac{1}{\sqrt{n + 2}}] \\ = & \lim_{n \to \infty} [\frac{1}{\sqrt{4}} - \frac{1}{\sqrt{n + 2}}] \\ = & \lim_{n \to \infty} [\frac{1}{2} - \frac{1}{\sqrt{n + 2}}] \\ = & \frac{1}{2} - \lim_{n \to \infty} \frac{1}{\sqrt{n + 2}} \\ = & \frac{1}{2} \end{array}$

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