Q. 54 Let a聽:聽[1,鈭�)鈫掆劃聽be a co... [FREE SOLUTION]

Chapter 7: Q. 54 (page 626)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information is:

${鈭�}_{1}^{鈭�} a (x) dx converges$ and

$a : [1, 鈭�) 鈫� 鈩� is continuous, decreasing and positive function .$

Step 2. Evaluating integral

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

Step 3. Solving for partial sum

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

Step 4. Result

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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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91影视

Let a:[1,鈭�)鈫�鈩�be a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral 鈭�1鈭�a(x)dxconverges, then the series localid="1649180069308" 鈭�k=1鈭�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

Decision Maths

Calculus

Applied Mathematics

Mechanics Maths

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.