Chapter 7: Q. 23 (page 657)

Check the convergence $\sum_{k = 1}^{\infty} \frac{k}{1 + k^{2}}$

Short Answer

Expert verified

Diverges

Step by step solution

Step 1. Given

The given series $\sum_{k = 1}^{\infty} \frac{k}{1 + k^{2}}$

Step 2. Limit comparison test

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

Step 3. Checking the convergence

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

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Check the convergence $\sum_{k = 1}^{\infty} \frac{k}{1 + k^{2}}$

Short Answer

Step by step solution

Step 1. Given

Step 2. Limit comparison test

Step 3. Checking the convergence

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Check the convergence∑k=1∞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

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Check the convergence $\sum_{k = 1}^{\infty} \frac{k}{1 + k^{2}}$