Chapter 7: Q. 15 (page 624)

Find an example of a continuous function f : $[1, \infty) \to R$ such that $\int_{1}^{\infty} f (x) d x$ diverges and localid="1649077247585" $\sum_{k = 1}^{\infty} f (k)$ converges.

Short Answer

Expert verified

An example is $\sin (πx)$ .

Step by step solution

Step 1. Given information.

Consider the given question,

The function is $[1, \infty) \to R$ .

The given integral is $\int_{1}^{\infty} f (x) d x$ .

Step 2. Evaluate the integral.

Evaluating the integral,

$\int_{1}^{\infty} f (x) d x = \int_{1}^{\infty} \sin (πx) d x = \lim_{k \to \infty} \int_{1}^{k} \sin (πx) d x = \lim_{k \to \infty} \int_{1}^{k} {[\frac{- \cos (πx)}{π}]}^{k}_{1} = \lim_{k \to \infty} \int_{1}^{k} [\frac{- \cos (πx) + \cos π}{π}] = \lim_{k \to \infty} \int_{1}^{k} [\frac{- \cos (πx) - 1}{π}] = \infty$

Thus, the improper integral $\int_{1}^{\infty} \sin (πx) d x$ is divergent.

Step 3. To prove the series as convergent.

The value of $\sum_{k = 1}^{\infty} f (k) = \sum_{k = 1}^{\infty} \sin (π k)$ is given below,

$\sum_{k = 1}^{\infty} \sin πk = \sin π + \sin 2 π + \sin 3 π + . . . \sum_{k = 1}^{\infty} \sin πk = 0 + 0 + 0 + . . . \sum_{k = 1}^{\infty} \sin πk = = 0$

Thus, the series $\sum_{k = 1}^{\infty} f (k) = \sum_{k = 1}^{\infty} \sin (π k)$ is convergent.

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Find an example of a continuous function f : $[1, \infty) \to R$ such that $\int_{1}^{\infty} f (x) d x$ diverges and localid="1649077247585" $\sum_{k = 1}^{\infty} f (k)$ converges.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Evaluate the integral.

Step 3. To prove the series as convergent.

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Find an example of a continuous function f :[1,∞)→Rsuch that ∫1∞fxdxdiverges and localid="1649077247585" ∑k=1∞fkconverges.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Evaluate the integral.

Step 3. To prove the series as convergent.

One App. One Place for Learning.

Most popular questions from this chapter

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Find an example of a continuous function f : $[1, \infty) \to R$ such that $\int_{1}^{\infty} f (x) d x$ diverges and localid="1649077247585" $\sum_{k = 1}^{\infty} f (k)$ converges.