the summation of series

gourish · 2014-04-19 18:26:51

i find it easy to find the sum of series of numbers raised to powers but when i need to find the sum of series like 1/n or 1/n^2 and so on it becomes tough i googled it but didn't find any satisfactory way to find such sum of series. so can anybody help me and please explain how to find those formula i don't like a list of formula which i will never remember

bobbym · 2014-04-19 20:38:16

Hi;

find the sum of series like 1/n or 1/n^2

You should never describe a sum without stating the indices of summation.

There are various proofs of that.

Has been known since the time of Euler and is done using Fourier series.

For other upper bounds than infinity there are additional methods.

i googled it but didn't find any satisfactory way to find such sum of series

Summation is considered to be tougher than integration and there is a branch of math called the summation calculus to deal with it. It requires a knowledge of differences. There are also numerical ideas, and tables are often used.

gourish · 2014-04-19 21:30:14

but what if i just to find the sum of the series from 1 to n where n belongs to natural numbers... it's basically from a series summation problem.... where the limits are not infinity but some finite value n but thanks for the help can you help me find that sum of 1/i and 1/i^2 for
i=1 to n

bobbym · 2014-04-19 22:50:42

Hi

Like in integration most sums can not be done in closed form so for this one they just invented a new function Hn, it is called the harmonic number.

Same thing for 1 / i^2.

gourish · 2014-04-20 00:16:34

can you elaborate on harmonic number i went through Wikipedia it's just mind boggling is there any simple way to use this harmonic number to find the sum of such series?

bobbym · 2014-04-20 00:33:16

That is the sum of the series.

that is how it is defined. Remember, there is no closed form known so they just went ahead and defined it as above.

gourish · 2014-04-20 00:36:22

so you still end up having another function but no actual answer in terms of numbers?

bobbym · 2014-04-20 00:41:06

You can estimate the answer to

using the methods of numerical analysis and thats where all this really belongs. But unless you add them all up manually you can not get the exact answer.

gourish · 2014-04-20 00:43:10

okay so i still need to use my calculator to end up with an answer

bobbym · 2014-04-20 00:48:12

Better yet, you will need your computer.

If we use numerical analysis we can estimate the sum to as many digits of precision as we require.

gourish · 2014-04-20 00:50:11

ok cool thanks

bobbym · 2014-04-20 01:01:02

For instance we can make use of this

to get a very good estimate to

The exact answer is

using 1) we get:

that is pretty close and we can do better...

Math Is Fun Forum

#1 2014-04-19 18:26:51

the summation of series

#2 2014-04-19 20:38:16

Re: the summation of series

#3 2014-04-19 21:30:14

Re: the summation of series

#4 2014-04-19 22:50:42

Re: the summation of series

#5 2014-04-20 00:16:34

Re: the summation of series

#6 2014-04-20 00:33:16

Re: the summation of series

#7 2014-04-20 00:36:22

Re: the summation of series

#8 2014-04-20 00:41:06

Re: the summation of series

#9 2014-04-20 00:43:10

Re: the summation of series

#10 2014-04-20 00:48:12

Re: the summation of series

#11 2014-04-20 00:50:11

Re: the summation of series

#12 2014-04-20 01:01:02

Re: the summation of series

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