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Problem 4

# Let $$a_{n}>0$$, and suppose $$\sum_{n=1}^{\infty} a_{n}$$ converges. If $$r_{n}=\sum_{k=n}^{\infty} a_{k}$$, show that (a) $$\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}$$ diverges (b) $$\sum_{n=1}^{\infty} \frac{a_{n}}{\sqrt{r_{n}}}$$ converges.

Expert verified
(a) The series $$\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}$$ diverges according to the limit comparison test. (b) The series $$\sum_{n=1}^{\infty} \frac{a_{n}}{\sqrt{r_{n}}}$$ converges also according to the limit comparison test.
See the step by step solution

## Step 1: Analyzing the provided series

Given $$a_{n}>0$$ and the series $$\sum_{n=1}^{\infty} a_{n}$$ converges, which implies the terms $$a_{n}$$ go to zero as $$n\to\infty$$. We also have $$r_{n}=\sum_{k=n}^{\infty} a_{k}$$, known as the 'tail' of the sequence starting at $$n$$, and clearly $$r_{n} \geq a_{n}$$ for all $$n$$, since each $$r_{n}$$ is a sum of terms, all of which are positive.

## Step 2: Application of the limit comparison test for (a)

We want to show $$\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}$$ diverges. Consider an easier series $$\sum_{n=1}^{\infty} \frac{a_{n}}{a_{n}} = \sum_{n=1}^{\infty} 1$$ which diverges. We have the ratio of corresponding terms of the two series as $$L=\lim_{n\to\infty} \frac{a_{n}/r_{n}}{a_{n}/a_{n}} = \lim_{n\to\infty} \frac{a_n}{a_n}.\frac{a_n}{r_n} = \lim_{n\to\infty} \frac{a_n}{r_n}$$. If this limit is positive and finite, by the limit comparison test, then the series $$\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}$$ also diverges. But this limit is equal to 1 since $$a_n < r_n$$ for all $$n$$. Therefore by limit comparison test, the series $$\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}$$ diverges.

## Step 3: Application of the limit comparison test for (b)

We want to show $$\sum_{n=1}^{\infty} \frac{a_{n}}{\sqrt{r_{n}}}$$ converges. Consider an easier series $$\sum_{n=1}^{\infty} \frac{a_{n}}{a_{n}^{1/2}} = \sum_{n=1}^{\infty} a_{n}^{1/2}$$ which converges (since $$\sum_{n=1}^{\infty} a_{n}$$ converges and $$a_{n}$$ go to zero as $$n\to\infty$$). Similar to the second step, we have the ratio of corresponding terms of the two series as $$L=\lim_{n\to\infty} \frac{a_{n}/\sqrt{r_{n}} }{a_{n}^{1/2} } =\lim_{n\to\infty} \frac{a_{n}^{1/2}.\sqrt{a_n}}{\sqrt{r_{n}} }=\lim_{n\to\infty} \frac{a_{n}}{\sqrt{r_{n}}}$$. If this limit is positive and finite, by the limit comparison test, then both series converge or both diverge. But this limit is equal to 1 since $$a_n < r_n$$ for all $$n$$. Therefore by the limit comparison test, the series $$\sum_{n=1}^{\infty} \frac{a_{n}}{\sqrt{r_{n}}}$$ converges.

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