Chapter 7: Q. 54 (page 626)

Let $a : [1, \infty) \to ℝ$ be a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral $\int_{1}^{\infty} a (x) d x$ converges, then the series localid="1649180069308" $\sum_{k = 1}^{\infty} a (k)$ does too.

Short Answer

Expert verified

$The sequence of partial sums s_{n} = \sum_{i = 1}^{\infty} a_{i} is bounded above by M . The terms are positive . Therefore, s_{n} \leq s_{n} + a_{n + 1} s_{n} \leq s_{n + 1} The sequence of partial sum is an increasing sequence and is bounded above by M . Therefore, the sequence {s_{n}} is convergent . Hence, the series \sum_{k = 1}^{\infty} a (k) is convergent .$

Step by step solution

Step 1. Given information is:

$\int_{1}^{\infty} a (x) dx converges$ and

$a : [1, \infty) \to ℝ is continuous, decreasing and positive function .$

Step 2. Evaluating integral

$The integral \int_{1}^{\infty} a (x) dx is convergent . If the integral is convergent, then the integral \int_{1}^{\infty} a (x) dx must have a finite value . Therefore, \int_{1}^{\infty} a (x) dx = M$

Step 3. Solving for partial sum

$It is given that the function a (x) is positive . Therefore, \int_{1}^{n} a (x) dx < \int_{1}^{\infty} a (x) dx . . . . . (1) Also, \sum_{i = 2}^{\infty} a_{i} < \int_{1}^{n} a (x) dx . . . . . (2) From equations (1) and (2), \sum_{i = 2}^{\infty} a_{i} < \int_{1}^{\infty} a (x) dx . . . . . (3) Also, \sum_{i = 1}^{\infty} a_{i} = a_{1} + \sum_{i = 2}^{\infty} a_{i} Using equation (3), \sum_{i = 1}^{\infty} a_{i} < \int_{1}^{\infty} a (x) dx \sum_{i = 1}^{\infty} a_{i} < M$

Step 4. Result

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Let $a : [1, \infty) \to ℝ$ be a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral $\int_{1}^{\infty} a (x) d x$ converges, then the series localid="1649180069308" $\sum_{k = 1}^{\infty} a (k)$ does too.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Evaluating integral

Step 3. Solving for partial sum

Step 4. Result

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Let a:[1,∞)→ℝbe a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral ∫1∞a(x)dxconverges, then the series localid="1649180069308" ∑k=1∞a(k)does too.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Evaluating integral

Step 3. Solving for partial sum

Step 4. Result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Let $a : [1, \infty) \to ℝ$ be a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral $\int_{1}^{\infty} a (x) d x$ converges, then the series localid="1649180069308" $\sum_{k = 1}^{\infty} a (k)$ does too.