Q. 23 Check the convergence聽鈭慿=1鈭�... [FREE SOLUTION]

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Chapter 7: Q. 23 (page 657)

Check the convergence ${鈭�}_{k = 1}^{鈭�} \frac{k}{1 + k^{2}}$

Short Answer

Expert verified

Diverges

Step by step solution

Step 1. Given

The given series ${鈭�}_{k = 1}^{鈭�} \frac{k}{1 + k^{2}}$

Step 2. Limit comparison test

$According to limit comparison test, let {鈭�}_{k = 1}^{鈭�} a_{k} a n d {鈭�}_{k = 1}^{鈭�} b_{k} be two series with positive terms 1 . If \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = L, where L is any positive real number, then either both series converges or both series diverges . 2 . \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0, {鈭�}_{k = 1}^{鈭�} b_{k} converges then {鈭�}_{k = 1}^{鈭�} a_{k} diverges 3 . \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭�, {鈭�}_{k = 1}^{鈭�} b_{k} diverges then {鈭�}_{k = 1}^{鈭�} a_{k} Converges$

Step 3. Checking the convergence

$a_{k} = \frac{1}{k + k^{2}} a n d b_{k} = \frac{1}{k} \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = \lim_{k 鈫� 鈭�} \frac{\frac{1}{k + k^{2}}}{\frac{1}{k}} s olving this limit = \frac{1}{1 + 0} = 1 {鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k} The given serie s {鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k} is a harmonic series, hence diverges From limit comparison test the series Diverges$ \

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91影视

Check the convergence ${鈭�}_{k = 1}^{鈭�} \frac{k}{1 + k^{2}}$

Short Answer

Step by step solution

Step 1. Given

Step 2. Limit comparison test

Step 3. Checking the convergence

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91影视

Check the convergence鈭�k=1鈭�k1+k2

Short Answer

Step by step solution

Step 1. Given

Step 2. Limit comparison test

Step 3. Checking the convergence

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Check the convergence ${鈭�}_{k = 1}^{鈭�} \frac{k}{1 + k^{2}}$