Math Is Fun Forum

ktesla39

Member

From: Nepal

Registered: 2024-10-08

Posts: 308

Website

Sum of infinite series.....

Hi,

Consider a series

which starts from 1 and ends up at infinity.

To find the sum of the series, we have to use

And if we combine it in a formula, we have

But can we factorise or make a simple equation to find

? I mean instead of finding all numbers and adding them, we should make  formula or any equation. I did try a lot but could not get any way. Can I get any help?

Last edited by ktesla39 (2025-04-22 23:59:40)

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"Talent hits the target no one else can hit. Genius hits the target no one else can see." - Arthur Schopenhauer

Bob

Administrator

Registered: 2010-06-20

Posts: 10,753

Re: Sum of infinite series.....

There is a formula for the sum of n terms.  The sum to infinity will be infinite so the formula for that is

Do you know the finite sum formula? Would you like me to derive it?

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ktesla39

Member

From: Nepal

Registered: 2024-10-08

Posts: 308

Website

Re: Sum of infinite series.....

Yes I wanted the finite formula. Like if we want to calculate S5, we will have to add 1+4+9+16+25 but we cant use this if we want to calculate large sum like S100. I need the formula because I am trying to make a formula to find the sum of quadratic equations like an^2 + bn + c .

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"Talent hits the target no one else can hit. Genius hits the target no one else can see." - Arthur Schopenhauer

Bob

Administrator

Registered: 2010-06-20

Posts: 10,753

Re: Sum of infinite series.....

So write out this expansion for n = 1 n=2 ...  n and sum them

The two cube sums have most terms in common so these can be cancelled. Also substitute the formula for the sum of n numbers

times by 2 to remove the fraction and expand the terms

Rearrange to make 6 times the required sum come onto the LHS and simplify the RHS

so the formula is:

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ktesla39

Member

From: Nepal

Registered: 2024-10-08

Posts: 308

Website

Re: Sum of infinite series.....

Thanks Bob, thanks a lot!

By using the formula you provided and the formula I previously made, I dedrived a new formula for sum of  quadratic series:

.

The formula is:

This might not be a lot useful, but I'm happy that I could do that. I checked it and it was working properly. Thanks Bob, thanks again!

Btw MIF is not responding to my emails. Will you try to contact him?

Last edited by ktesla39 (2025-04-24 01:18:29)

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"Talent hits the target no one else can hit. Genius hits the target no one else can see." - Arthur Schopenhauer

Bob

Administrator

Registered: 2010-06-20

Posts: 10,753

Re: Sum of infinite series.....

hi ktesla39

Using the email you registered with, my PA, Matty, has sent you a message. Did you spot it?

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ktesla39

Member

From: Nepal

Registered: 2024-10-08

Posts: 308

Website

Re: Sum of infinite series.....

Hi Bob,
I was at school. Just came back. I turned on my laptop and I got that.

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"Talent hits the target no one else can hit. Genius hits the target no one else can see." - Arthur Schopenhauer