Summation of Series -2

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Summation of Series -2

Mathematics , Integral Calculus

Summation of Series - 2

The method of differences

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

$Delta t_1,,Delta t_2,,Delta t_3,,ldots$

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

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

Repeating the procedure, we can obtain the series of various order of differences:

$egin{array}{llllllllllll} t_1 & & t_2 & & t_3 & & t_4 & & t_5 & & t_6 & ldots \ & Delta t_1 & & Delta t_2 & & Delta t_3 & & Delta t_4 & & Delta t_5 & & ldots & & & Delta_2 t_1 & & Delta_2 t_2 & & Delta_2 t_3 & & Delta_2 t_4 & & ldots & & & & & Delta_3 t_1 & & Delta_3 t_2 & & Delta_3 t_3 & & ldots & & & & & & & ldots & ldots & ldots & ldots & ldots end{array}$

There exists two cases in which we can easily find the $n$ -th term and the sum up to the $n$ -th term of the series.

i) While formning the series of various order of differences, we eventually come to a series in which all terms are equal. (This will always happen if the $n$ -th term of the series will be a polynomial in $n$ .)

Let us denote the first term of the given series by $a$ , the first term of the series of the first order differences by $d_1$ , the first term of the series of the second order differences by $d_2$ , the first term of the series of the third order differences by $d_3$ , and so on. That is, let $a=t_1$ , $d_1 = Delta t_1$ , $d_2 = Delta_2 t_1$ , $d_3 = Delta_3 t_1$ , and so on. Then,

$t_n = a + (n-1)d_1 + rac{(n-1)(n-2)}{2!}d_2 + rac{(n-1)(n-2)(n-3)}{3!}d_3 + ldots$

and the sum upto $n$ terms is

$S_n = na + rac{n(n-1)}{2!}d_1 + rac{n(n-1)(n-2)}{3!}d_2 + rac{n(n-1)(n-2)(n-3)}{4!}d_3 + ldots$

For example, consider the series

$12,,40,,90,,168,,280,,432,,ldots$

We consider the series and various successive orders of difference

$egin{array}{llllllllllll} 12 & & 40 & & 90 & & 168 & & 280 & & 432 & ldots \ & 28 & & 50 & & 78 & & 112 & & 152 & & ldots \ & & 22 & & 28 & & 34 & & 40 & & ldots \ & & & 6 & & 6 & & 6 & & ldots\ & & & & 0& & 0 & & 0 & & ldots end{array}$

So by using the above rule, we obtain the $n$ -th term

$t_n = 12 + 28(n-1) + rac{22(n-1)(n-2)}{2!} + rac{6(n-1)(n-2)(n-3)}{3!}$

$=n^3 + 5n^2 + 6n$

And the sum

$S_n = 12n+ rac{28n(n-1)}{2!}+ rac{22n(n-1)(n-2)}{3!}+ rac{6n(n-1)(n-2)(n-3)}{4!}$

$= rac{1}{12}n(n+1)(3n^2+23n+46)$

ii) While formning the series of various order of differences, we eventually come to a series which is a GP.

Suppose the $p$ -th order of differences of a series foms a GP with common ratio $r$ . In this case, we can assume the $n$ -th term as

$t_n = a r^{n-1} + arphi(n)$ ,

where $a$ is a constant to be determined, and $arphi(n)$ is a polynomial in $n$ with degree not exceeding $p-1$ .

For example, suppose we need to find the sum upto $n$ terms of the series

$10,,23,,60,,169,,494,,ldots$

We first find various order differences.

$egin{array}{llllllllllll} 10 & & 23 & & 60 & & 169 & & 494 & & ldots & ldots \ & 13 & & 37 & & 109 & & 335 & & & & ldots \ & & 24 & & 72 & & 216 & & ldots & & ldots end{array}$

So, we see that the 2nd order differences are in GP with common ratio 3. Accordingly, we assume

$t_n = a 3^{n-1} + b n + c$

To determine the constants $a$ , $b$ , and $c$ , make $n$ equal to 1, 2, 3, successively; then

$a+b+c=10,quad 3a + 2b + c = 23, quad 9a + 3b +c = 60$

from where we obtain $a=6$ , $b=1$ , $c=3$ . Hence,

$t_n = 6cdot 3^{n-1} + n +3= 2cdot 3^n + n + 3$

Now, we can easily detrmine the sum up to $n$ terms as

$S_n = 6 cdot rac{3^n-1}{3-1} + rac{n(n+1)}{2} + 3n = 3 (3^n-1)+ rac{n(n+7)}{2}$

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Comments (14)

Sahil Gupta

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30 Sep 2008 02:03:17 IST

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sir you are a genious

Sahil Gupta

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30 Sep 2008 02:08:06 IST

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sorry genius

svj 29

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30 Sep 2008 15:55:03 IST

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awesome...!!!

vicor

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30 Sep 2008 19:46:25 IST

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wow...........!!!!!!!!!!

Rajat Sen

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30 Sep 2008 20:37:12 IST

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very useful

Yash: Save Tigers..!!

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30 Sep 2008 22:18:50 IST

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awesum!~

®µD®A

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30 Sep 2008 22:28:12 IST

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these articles are meant to printed out and read every day at the morning...

Great and awesome as the previous one!

Mirka

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2 Oct 2008 16:32:46 IST

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Really Awesome!!!
... True Genius

puja singh

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3 Oct 2008 14:46:54 IST

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2ruly awesummmmmmmmm

hats off

kanika goel

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18 Nov 2008 22:12:48 IST

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thanks a lot for helping us sir!!

livehappily livehappily

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21 Nov 2008 07:43:47 IST

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gr8 job

taran

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8 Mar 2009 00:16:02 IST

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wowwwwwwww!!!!!!!!!!!!!!!!!!thank u very much sir

eragon24

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31 Aug 2009 13:18:40 IST

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rekindling it for everyones benifit

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24 Dec 2009 16:52:45 IST

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thnx sir :)

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