Suppose that ak 0. Prove that k=1 ak converges if and only if the series k=1

Question:

Suppose that ak → 0. Prove that ∑∞k=1 ak converges if and only if the series ∑∞k=1 (a2k + a2k+1) 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 04:06 AM

See More Questions

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...
Suppose that ak > 0 and that ak1/k as k . Prove that k=1 akxk converges absolutely for all |x| < 1/a if a 0 and for all x R if a = 0.
Suppose that ak I 0 as k . Prove that k=1 ak sin kx converges uniformly on any closed interval [a, b] (0, 2).
Kim Corporation, a calendar year taxpayer, has manufacturing facilities in States A and B. A summary of Kims property holdings follows. Determine Kims property factors for the two states assuming...
Assume a firm has several hundred possible investments and that it wants to analyze the risk-return trade-off for portfolios of 20 projects. How should it proceed with the evaluation?
Write each percent as a fraction. Give answers in lowest terms. 180%
A food processing company is designing the packaging for a new product line of potato chips. The expected shelf life of the potato chips is 10 months. The bag has to hold $200 \mathrm{~g}$ of...
Crest Industries sells a single model of satellite radio receivers for use in the home. The radios have the following price and cost characteristics: Sales price . . . . . . . . . . . $ 80 per radio...
Mountain Dew Partnership (MDP) was setup 8 years ago. It was formed by 2 brothers Tawanda and Tendai and their best friend Brian. Tawanda and Tendai are originally from Zimbabwe but are now British...
Travel Warehouse distributes suitcases to retail stores and extends credit terms of n/30 to all of its customers. Travel Warehouse uses a perpetual inventory system and at the end of June its...
Suppose that a,b R satisfy b/a RZ. Find all q > 0 such that converges. ak +bq k=1
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) Suppose that 0 b) If k=1 ak...
A psychology class consists of 32 freshmen and 48 sophomores. If the professor selects names from the class list using random sampling, a. What is the probability that the first student selected will...
A horse pulls a plow a distance of 29 m along the ground from (0, 0) to (29, 0), during which a force of 750 N is applied at an angle of 10 with respect to the ground. What is the work done by the...
While you are riding the merry-go-round with your friends, you start dancing in the middle of the merry-go-round. One of your friends tries standing up at the edge of the ride and is thrown off of...
A ball is released with some initial velocity, and accelerates under free-fall towards earth. After falling 61.2 meters, the velocity of the ball is 40.1 m/s. Calculate the magnitude of the initial...
A speeding person in a car passes a police car, which is parked on the side of the road, at 1 0 0 km / hand continues at constant speed. After 2 s ( seconds ) the police car takes up the pursuit of...
Consider a penstock that is vertical and has a height of h = 8 5 . 5 = 8 5 . 5 m . . How long, in seconds, does it take water to fall from the top of the penstock to the bottom? Assume the water...
On May 1, 202X, Magdalena Company received a $30,000, 90-day, 9% note from Quincy Company dated May 1. On June 20, 202X, Magdalena discounted the note at Pittsfield Bank at a discount rate of 12%. 1....
For each equation, (a) Write it in slope-intercept form (b) Give the slope of the line (c) Give the y-intercept (d) Graph the line. 7x - 3y = 3
The permeability of a membrane used as a moisture barrier in a biological application depends on the thickness of three integrated layers. Layers 1, 2, and 3 are normally distributed with means of...
A small company is to decide what investments to use for cash generated from operations. Each investment has a mean and standard deviation associated with the percentage gain. The first security has...
Eight measurements were made on the inside diameter of forged piston rings used in an automobile engine. The data (in millimeters) are 74.001, 74.003, 74.015, 74.000, 74.005, 74.002, 74.005, and...
The following partially complete T-accounts for the month of June along with additional information are from Renfrew & Company: Materials Inventory Work-in-Process Inventory Finished Goods Inventory...
The information that follows pertains to Julia Company: (a) Temporary differences for the year 2024 are summarized below. Expenses deducted in the tax return, but not included in the income...
Harwood Company uses a job-order costing system that applies overhead cost to jobs on the basis of machine-hours. The company's predetermined overhead rate of $2.60 per machine-hour was based on a...

Suppose that ak 0. Prove that k=1 ak converges if and only if the series k=1

Question:

Step by Step Answer:

If sn n k1 a k converges then so does s ...View the full answer

An Introduction to Analysis

Students also viewed these Numerical Analysis questions