Proof of alternating series test, Mathematics

Assignment Help:

Proof of Alternating Series Test

With no loss of generality we can assume that the series begins at n =1. If not we could change the proof below to meet the new starting place or we could perform an index shift to obtain the series to begin at n =1 .

First, notice that because the terms of the sequence are decreasing for any two successive terms we can say,

bn - bn+1 ≥ 0

Here now, let us take a look at the even partial sums.

s2 = b1 - b2 ≥ 0

s4 = b1 - b2 + b3 - b4 = s2 + b3 - b4 ≥ s2                                              because b3 - b4 > 0

S6 = s4 + b5 - b6  ≥ s4                                                            because b5 - b6 > 0     

S2n = S2n -2 + b2n -1 - b2n  ≥ S2n -2                                                           because b2n-1 - b2n > 0

Thus, {S2n}is an increasing sequence.

 Next, we can as well write the general term as,

S2n = b1-b2 + b3 - b4 + b5 + .... - b2n-2 + b2n-1 - b2n

= b1 - (b2-b3) - (b4 - b5) + ..... - (b2n-2 - b2n-1) - b2n

Every quantity in parenthesis is positive and by assumption we be familiar with that b2n is as well positive.  Thus, this tells us that S2n< b1 for all n.

We now be familiar with that {S2n}is an increasing sequence that is bounded above and thus we know that it must as well converge.  Thus, let's assume that its limit is s or,

1578_Proof of Alternating Series Test 1.png

Subsequently, we can quickly find out the limit of the sequence of odd partial sums, {S2n+1} as follows,

1043_Proof of Alternating Series Test 2.png

Thus, we now know that both of the {S2n} and {S2n+1} are convergent sequences and they both have similar limit and so we as well know that {Sn} is a convergent sequence along with a limit of s.  This in turn tells us that ∑an is convergent.


Related Discussions:- Proof of alternating series test

Devision, how many times can u put 10000 into 999999

how many times can u put 10000 into 999999

Calculate moving average, Calculate Moving Average The table given bel...

Calculate Moving Average The table given below represents company sales; calculate 3 and 6 monthly moving averages, for data Months Sales

Prove that seca+tana=2x, If secA= x+1/4x, prove that secA+tanA=2x or  1/2x....

If secA= x+1/4x, prove that secA+tanA=2x or  1/2x. Ans:    Sec? = x +  1/4x ⇒ Sec 2 ? =( x + 1/4x) 2                             (Sec 2 ?= 1 + Tan 2 ?) Tan 2 ? = ( x +

Algorithm for division helping a child grasp, E1) Why don't you think of so...

E1) Why don't you think of some activities for the same purpose now? E2) Suggest, in detail, another activity for helping a child grasp the algorithm for division. We come to

Integrals involving quadratics - integration techniques, Integrals Involvin...

Integrals Involving Quadratics To this point we have seen quite some integrals which involve quadratics.  Example of Integrals Involving Quadratics is as follow: ∫ (x / x 2

Arthemetic progreession, ball are arranged in rows to form an equilateral t...

ball are arranged in rows to form an equilateral triangle .the firs row consists of one abll,the second of two balls,and so on.If 669 more balls are added,then all the balls canbe

Problem solving involving quadratic equations, a painting is 20 cm wider th...

a painting is 20 cm wider than its height. its area is 2400 centimeter squared. find its lenght and width

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd