[ABEL'S TEST] Suppose that k=1 ak converges and that bk b as k . Prove

Question:

[ABEL'S TEST] Suppose that ∑∞k=1 ak converges and that bk ↓ b as k → ∞. Prove that ∑∞k=1 akbk converges.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

An Introduction to Analysis

ISBN: 978-0132296380

4th edition

Authors: William R. Wade

See More Books

Question Posted: Mar 29, 2016 04:07 AM

See More Questions

Suppose that [ak] is a sequence of nonzero real numbers and that exists as an extended real number. Prove that k=1 ak converges absolutely when p > 1. ak+1 (1- p= lim k (1- ak
Let {ak} and be real sequences. Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones. a) If ak is strictly decreasing...
Let {ak} and {bk} be real sequences. Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones. a) If k=1 ak converges and...
The average of 4 consecutive odd positive integers is 16. The product of the smallest and largest positive odd integer is 1. 210 2. 247 3. 294 4. 320 None of the above 5.
Davis Chili Company is considering an investment of $35,000, which produces the following inflows: YearCash Flow 1 .......$16,000 2 ....... 15,000 3 ....... 12,000 You are going to use the net...
As a summer intern in the Marketing Department at Garrison Scientific, Inc., in Waltham, Massachusetts, you have been working on the companys annual catalog. You notice that staffers could save a lot...
Consider the gasoline mileage data in Table B.3. a. Fit a multiple linear regression model relatmg gasoline mileage $y$ (miles per gallon) to engine displacement $x_{1}$ and the number of carburetor...
Al Rosen invests $25,000 in a mint condition 1952 Mickey Mantle Topps baseball card. He expects the card to increase in value 12 percent per year for the next 10 years. How much will his card be...
Consider three incomes: $35 000, $65 000, and $130 000. Determine the maximum RRSP contribution and the tax benefit at each income level. What would be an equivalent TFSA contribution? Next project...
Why do mitochondria need a special translocator to import proteins across the outer membrane, when the membrane already has large pores formed by porins?
Using any test covered in this chapter so far, find out which of the following series converge absolutely, which converge conditionally, and which diverge. a) b) c) d) e) (-k' i " (-1)(-3) 1-4 (1-2k...
Show that under the hypotheses of Dirichlet's Test, Sk (bx-b+ agb =sk (bx- br+1). k=1 k=1
What is the role of a WTA server? What are the different ways of integrating WTA servers into the WAP architecture?
Calculate the price of a six-month European put option on the spot value of an index. The six-month forward price of the index is 1,400, the strike price is 1,450, the risk-free rate is 5%, and the...
Xanadu Mining is considering three mutually exclusive alternatives, as shown in the table below. MARR is 10 percent/year. a. What is the present worth of each alternative? b. Which alternative should...
What is the formula for the delta of a call option on a stock that provides no income?
The present worth of a multiyear investment with all positive cash flows (incomes) other than the initial investment is $P W=\$ 10,000$ at $M A R R=i \%$. If MARR changes to $(i+1) \%$, the...
Two numerically controlled drill presses are being considered by the production department of Zunnis Manufacturing; one must be selected. Comparison data is shown in the table below. MARR is 10...
The electric field intensity of a uniform plane wave in a medium ( = 0, = o , = o r ) is E = 12 sin(2 x 10 7 t 3y)a z V/m (a) Determine the polarization of the wave. (b) Find the frequency. (c)...
In Exercises 15 through 30, find the derivative dy/dx. In some of these problems, you may need to use implicit differentiation or logarithmic differentiation. y ex + et -2x 1 + e
Reconsider the spot weld shear strength data in Exercise 6-23. Construct a box plot of the strengths and write an interpretation of the plot. How does the box plot compare in interpretive value to...
Reconsider the female engineering student height data in Exercise 6-22. Construct a box plot of the heights and write an interpretation of the plot. How does the box plot compare in interpretive...
Reconsider the water quality data in Exercise 6-24. Construct a box plot of the concentrations and write an interpretation of the plot. How does the box plot compare in interpretive value to the...
Tim Drinking, Inc. used a predetermined overhead allocation rate to allocate of indirect costs to the Filling Department and the Packaging Department journal entry to record the allocation of...
What are the key principles of management, and how do they contribute to effective organizational practices?
Discuss the impact of globalization on small businesses in developing countries. Analyze the advantages and disadvantages of globalization for small businesses, and provide recommendations for...

[ABEL'S TEST] Suppose that k=1 ak converges and that bk b as k . Prove

Question:

Step by Step Answer:

If b k b then b k b 0 as k Moreover if k1 a k converges the...View the full answer

An Introduction to Analysis

Students also viewed these Numerical Analysis questions