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Chapter 7: Q. 37 (page 592)

Find the least upper bound of the sequences in Exercises 37–42
$\{2 - \frac{1}{k^{2}}\}$

Short Answer

Expert verified

The east upper bound is $2 .$

Step by step solution

Step 1. Given information.

The given series is $\{2 - \frac{1}{k^{2}}\}$ .

Step 2. Upper bound.

Now, we can write,

$\lim_{k \to \infty} a_{k} = \lim_{k \to \infty} (2 - \frac{1}{k^{2}}) = \lim_{k \to \infty} 2 - \lim_{k \to \infty} (\frac{1}{k^{2}}) = 2$

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Most popular questions from this chapter

Determine whether the series $\sum_{k = 0}^{\infty} \frac{4^{k + 1}}{3^{2 k}}$ converges or diverges. Give the sum of the convergent series.

$I f \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty and \sum_{k = 1}^{\infty}_____Converges the n \sum_{k = 1}^{\infty}_____(converges / diverges) .$

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.

Let $α$ be any real number. Show that there is a rearrangement of the terms of the alternating harmonic series that converges to $α$ . (Hint: Argue that if you add up some finite number of the terms of $\sum_{k = 1}^{\infty} \frac{1}{2 k - 1}$ , the sum will be greater than $α$ . Then argue that, by adding in some other finite number of the terms of

$\sum_{k = 1}^{\infty} (- \frac{1}{2 k})$ , you can get the sum to be less than $α$ . By alternately adding terms from these two divergent series as described in the preceding two steps, explain why the sequence of partial sums you are constructing will converge to $α$ .)

Determine whether the series $\sum_{k = 2}^{\infty} {(\frac{- 3}{5})}^{k}$ converges or diverges. Give the sum of the convergent series.

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Find the least upper bound of the sequences in Exercises 37–422-1k2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Upper bound.

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Find the least upper bound of the sequences in Exercises 37–42
$\{2 - \frac{1}{k^{2}}\}$