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Chapter 12: Problem 26

Determine whether the series converges or diverse. $$\sum \frac{2 k+1}{\sqrt{k^{3}+1}}$$

Short Answer

Expert verified
The series diverges.

Step by step solution

01

Identify Test

Because the series possesses both positive terms and an apparent end behavior, the Comparison Test seems applicable.
02

Identify Comparable series

Next is to find a series to compare. In this case, if you simplify the terms of the series by removing lower order terms, you get \( a_n = \frac{2n}{n^{3/2}} = \frac{2}{n^{1/2}} \). So it is cleared that the series \( \sum \frac{2}{n^{1/2}} \) can be a good series to compare the given series to.
03

Apply Comparison Test Criteria

For all \( k \geq 1 \) in the function, \( \frac{2k+1}{\sqrt{k^{3}+1}} \leq \frac{2k}{k^{3/2}} = \frac{2}{k^{1/2}} \) which means that for all \( k \geq 1 \), the terms of the given series are less than or equal to the terms in the series \( \sum \frac{2}{k^{1/2}} \) which is a p-series with \( p=1/2 \). Considering that p-series will only converge if \( p > 1 \) and here \( p=1/2<1 \). So, the comparison series diverges.
04

Conclude if Series Converges or Diverges

Since the comparison series diverges according to the Comparison Test, the original series \( \sum \frac{2k+1}{\sqrt{k^{3}+1}} \) also diverges.

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Most popular questions from this chapter

Show that if \(\sum a_{2}\) is absolutely convergent and \(\left|b_{k}\right| \leq\left|a_{k}\right|\) for at \(J k,\) then \(\sum b_{k}\) is absolutely convergent.

Derive a series expansion in \(x\) for the function and specify the numbers \(x\) for which the expansion is valid. Take \(a>0\). $$f(x)=\ln (a+x)$$

Take \(r>0\) and let the \(a_{k}\) be positive. Use the root test to show that, if \(\left(a_{k}\right)^{1 / k} \rightarrow \rho\) and \(\rho<1 / r,\) then \(\sum a_{i} r^{k}\) converges.

Determine whether the series converges or diverges. $$\frac{2}{3}+\frac{2 \cdot 4}{3 \cdot 7}+\frac{2 \cdot 4 \cdot 6}{3 \cdot 7 \cdot 11}+\frac{2 \cdot 4 \cdot 6 \cdot 8}{3 \cdot 7 \cdot 11 \cdot 15}+\cdots$$

Speed of convergence) Find the least integer N for which the \(n\)th partial sum of the series differs from the sum of the series by less than 0.0001. $$\sum_{k=0}^{\infty}(0.9)^{k}$$
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