Chapter 7: Q. 88 (page 616)

Let $\sum_{k = 1}^{\infty} a_{k} b e a c o n v e r g e n t s e r i e s a n d \sum_{k = 1}^{\infty} b_{k} b e a d i v e r g e n t s e r i e s .$ Prove that the series $\sum_{k = 1}^{\infty} (a_{k} + b_{k})$ diverges.

Short Answer

Expert verified

Proof by method of contradiction.

$\sum_{k = 1}^{\infty} (a_{k} + b_{k})$ is a divergent series.

Step by step solution

Step 1. Given Information.

$\sum_{k = 1}^{\infty} a_{k}$ is a convergent series and $\sum_{k = 1}^{\infty} b_{k}$ is a divergent series.

Step 2. Proof by method of contradiction.

Let suppose $\sum_{k = 1}^{\infty} (a_{k} + b_{k})$ be a convergent series.

Now, we know the sum of two convergent series is a convergent series.

So, $\sum_{k = 1}^{\infty} (a_{k} + b_{k}) - \sum_{k = 1}^{\infty} (a_{k})$ will also be convergent.

Which can be implied as $\sum_{k = 1}^{\infty} (a_{k} + b_{k} - a_{k}) = \sum_{k = 1}^{\infty} (b_{k})$ must be convergent.

But it is given that $\sum_{k = 1}^{\infty} (b_{k})$ is a divergent series.

So it contradicts and so our assumption is wrong.

Thus the series $\sum_{k = 1}^{\infty} (a_{k} + b_{k})$ is a divergent series.

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Let $\sum_{k = 1}^{\infty} a_{k} b e a c o n v e r g e n t s e r i e s a n d \sum_{k = 1}^{\infty} b_{k} b e a d i v e r g e n t s e r i e s .$ Prove that the series $\sum_{k = 1}^{\infty} (a_{k} + b_{k})$ diverges.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof by method of contradiction.

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Let ∑k=1∞akbeaconvergentseriesand∑k=1∞bkbeadivergentseries.Prove that the series ∑k=1∞ak+bkdiverges.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof by method of contradiction.

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Most popular questions from this chapter

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Let $\sum_{k = 1}^{\infty} a_{k} b e a c o n v e r g e n t s e r i e s a n d \sum_{k = 1}^{\infty} b_{k} b e a d i v e r g e n t s e r i e s .$ Prove that the series $\sum_{k = 1}^{\infty} (a_{k} + b_{k})$ diverges.