(a) Use the Integral Test to show that the series -1() converges. Hint: use integration by parts.(b) Find the value of the approximation s5 = Ek=1(k) to four decimal places.(c) The Remainder Estimate for the Integral Test states that the nth remainder R₂ satisfies+₁ f(x) dx ≤ R₂ ≤ f(x)dx.Use this, along with your work in part (a), to find lower and upper bounds on the 5th remainder R5 when wekuse 55 = x=1Give four decimal places.to approximate the value of the series s = Σk=1 ¹(3)k(d) Then use your work above, along with Inequality [3] in Section 11.3 of the text, to give a betterestimate for the value of the series s= Σ=1*Give four decimal places.1(3).(e) Find n sufficiently large so that our estimate in part (d) following the procedure above would beaccurate to within 0.005. What estimate for the series value does using this value of n give you?

Answered: Σ=1(*) (2) Use the Integral Test to…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

(a) Use the Integral Test to show that the series -1() converges. Hint: use integration by parts.
(b) Find the value of the approximation s5 = Ek=1(k) to four decimal places.
(c) The Remainder Estimate for the Integral Test states that the nth remainder R₂ satisfies
+₁ f(x) dx ≤ R₂ ≤ f(x)dx.
Use this, along with your work in part (a), to find lower and upper bounds on the 5th remainder R5 when we
k
use 55 = x=1
Give four decimal places.
to approximate the value of the series s = Σk=1 ¹(3)
k
(d) Then use your work above, along with Inequality [3] in Section 11.3 of the text, to give a better
estimate for the value of the series s
= Σ=1*
Give four decimal places.
1(3).
(e) Find n sufficiently large so that our estimate in part (d) following the procedure above would be
accurate to within 0.005. What estimate for the series value does using this value of n give you?

Expert Solution

Step 1

Since you have posted multiple questions, we will solve first question for you. If you want any specific question to be solved then please specify the question number or post only that question.

Consider the series $\sum_{k = 1}^{\infty} (\frac{k^{2}}{e^{k}})$ .

By using integral test to show that given series converges.

If the function $f (x)$ is positive, decreasing and continuous function on $[1, \infty)$ such that $f (k) = a_{k}$ then if $\int_{1}^{\infty} f (x) d x$ converges then the series $\sum_{k = 1}^{\infty} a_{k}$ converges.