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

Total accumulation of the amount deposited in saving account, A bank pays o...

A bank pays on its savings an interest rate of 6% per year but compounds interest monthly (i.e., estimates the interest each month and adds it to the balance).  You plan to deposit

Nine minus five times a number, Nine minus five times a number, x, is no le...

Nine minus five times a number, x, is no less than 39. Which of the subsequent expressions represents all the possible values of the number? Translate the sentence, "Nine minus

One integer is four times other what is the value of lesser, One integer is...

One integer is four times other. The sum of the integers is 5. What is the value of the lesser integer? Let x = the lesser integer and now let y = the greater integer. The ?rst

Assignment help job, Sir before I applied for online assignment help job an...

Sir before I applied for online assignment help job and the selection process is not complete for me. You sent me problem assignment before.But those problems were not completed.Ca

Perimeter of trinagle, what is the perimeter of a triangele with the sides ...

what is the perimeter of a triangele with the sides of 32 in /22 in/20 in/

Develop a linear program to investment advisory firm, An investment advisor...

An investment advisory firm manages funds for its numerous clients. The company uses an asset allocation model that recommends the portion of each client's portfolio to be invested

Estimate how much should every friend pay, A group of ?ve friends gone out ...

A group of ?ve friends gone out to lunch. The total bill for the lunch was $53.75. Their meals all cost about the similar, so they needed to split the bill evenly. Without consider

Write Your Message!

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