Summation of Series

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26 Sep 2008 07:41:52 IST

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26 Sep 2008 07:41:52 IST

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Summation of Series

Mathematics , Analytical Geomerty ,

Summation of Series

We already are familiar with the Arithmetic series, Geometric series, and the Harmonic series and the ways of finding their sums. For any arbitraray series, the general way to find the sum is to express the n-th term as the difference of two quantities one of which is the same function of n that the other is of n-1 i.e.

$t_n = f(n)-f(n-1)$

In this case, when one adds up the series upto the n terms, what remains is simply

$S_n = f(n)-f(0)$

But in certain cases, we can find the sum of n terms of the series by using the following some general rules, that I am giving below.
1. To find the sum of n terms of a series, each term of which is composed of r factors in A.P., the first factors of the terms being in the same A.P. : Write down the n^th term, affix the next factor at the end, divide by the number of factors thus increased and also by the common difference, and add a constant.

Suppose I need to find the sum of the first n terms of the series: $3cdot 7 cdot 11 cdot 15 + 7 cdot 11 cdot 15 cdot 19 + 11 cdot 15 cdot 19 cdot 13 + ldots$

We note that each term is consisting of four factors which are in AP and the first factors of each term are in the same AP 3, 7, 11, . . .. The common difference is 4. The n-th term of the series is easily seen to be $t_n = (4n-1)(4n+3)(4n+7)(4n+11)$
Therefore, using the above rule, we get ,

$S_n = rac{(4n-1)(4n+3)(4n+7)(4n+11)(4n+15)}{5cdot 4} + c$

To determine $c$ , set $n=1$ , then we get

$3cdot 7 cdot 11 cdot 15 = rac{3cdot 7 cdot 11 cdot 15cdot 19}{20}+c$

Therefore, $c= rac{693}{4}$

Hence, we get $S_n = rac{(4n-1)(4n+3)(4n+7)(4n+11)(4n+15)}{5cdot 4} + rac{693}{4}$

2. To find the sum of n terms of a series, each term of which is composed of the reciprocal of the product of r factors in A.P., the first factors of the terms being in the same A.P. : Write down the n^th term, strike off the factor from the begining, divide by the number of factors thus reduced and also by the common difference, change the sign and a constant.

Suppose we need to find the sum $rac{1}{1cdot 2cdot 3cdot 4} + rac{1}{2cdot 3cdot 4cdot 5} + rac{1}{ 3cdot 4cdot 5 cdot 6} + ldots$ . We note that the $n$ -th term is

$t_n= rac{1}{n(n+1)(n+2)(n+3)}$

So, using the above rule, the required sum

$S_n = C - rac{1}{3cdot 1cdot (n+1)(n+2)(n+3)}$

To find $C$ , set $n=1$ , then $rac{1}{1cdot 2cdot3cdot4}=C- rac{1}{3cdot 2cdot3cdot4}$ ; therfore $C= rac{1}{18}$ . Hence,

$S_n = rac{1}{18}- rac{1}{3 (n+1)(n+2)(n+3)}$

3. The method of differences - I. Suppose we have a series of numbers

$t_1,,t_2,,t_3,,t_4,, ldots$

Obtain a second series by subtracting each term from the term that immediately follows it. The series

$t_2-t_1,, t_3-t_2,, t_4-t_3, ldots$

thus formed is called the series of the first order of differences , and can be conveniently denoted by

$Delta_1 t_1, , Delta_1 t_2,, Delta_1 t_3,,ldots$

From this series, we can obtain the series of the second order differences by subtracting each term from the term following it, denote it by

$Delta_2 t_1, , Delta_2 t_2,, Delta_2 t_3,,ldots$

To be finished .......

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Comments (12)

>>>>><<<<<

Cool goIITian

Joined: 19 May 2008 13:34:52 IST

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26 Sep 2008 13:16:05 IST

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sir can we apply the same concept for normal ap gp hp pls contribute to this sir

Sahil Gupta

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27 Sep 2008 23:06:46 IST

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mind-blowing sir

®µD®A

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28 Sep 2008 00:24:36 IST

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speechless. Sir can you give some tips about complex numbers?

®µD®A

Blazing goIITian

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28 Sep 2008 09:32:32 IST

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Sir there is a typo that should be 23 rather that 13

Mirka

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29 Oct 2008 13:48:56 IST

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true genius!

spiderman

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18 Nov 2008 10:54:46 IST

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very interesting sir,please keep sending such articles!!!!!!!!!!!!

Pranav Pant

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18 Nov 2008 16:58:27 IST

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Awesome.

kanika goel

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18 Nov 2008 21:56:05 IST

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thanks a ton sir..
i used to adapt that long method earlier..
but this..is really...too usefull..
keep posting such articles..for our benefit!!

eragon24

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31 Aug 2009 13:19:11 IST

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rekindling it for everyones benifit

!!**~~AKRITI~~**!!

Hot goIITian

Joined: 27 May 2009 13:40:18 IST

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1 Sep 2009 06:51:37 IST

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itz really very useful n enlightening...thnx 4 such an article..!!thnx a million sir!:)

AIR 538 Asish

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24 Dec 2009 14:23:09 IST

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rekindled (again)

dc16

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24 Dec 2009 16:51:07 IST

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Great SiR ........

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