Closed Expression for Series

• Aug 31st 2011, 08:41 AM
veronicak5678
Closed Expression for Series
By E I mean sum from k=1 to n

Find a closed expression for

1+ E [1 / 2^(k-1)]

I used the formula for geo series and have the sum = 1 - 1/2^n. Is this correct? How do I make a closed expression?
• Aug 31st 2011, 10:05 AM
Prove It
Re: Closed Expression for Series
Quote:

Originally Posted by veronicak5678
By E I mean sum from k=1 to n

Find a closed expression for

1+ E [1 / 2^(k-1)]

I used the formula for geo series and have the sum = 1 - 1/2^n. Is this correct? How do I make a closed expression?

You have $\displaystyle \displaystyle 1 + \sum_{k = 1}^n{\frac{1}{2^{k-1}}} = 1 + \sum_{k = 1}^n{\left(\frac{1}{2}\right)^{k - 1}}$

This sum is an infinite geometric series with $\displaystyle \displaystyle a = 1, r = \frac{1}{2}$. Since $\displaystyle \displaystyle |r| < 1$, the sum is equal to $\displaystyle \displaystyle \frac{a}{1 - r}$. Go from here.
• Aug 31st 2011, 10:45 AM
veronicak5678
Re: Closed Expression for Series
I thought that formula was for infinite series only? Can I use it if I go from 1 to n?
• Aug 31st 2011, 10:54 AM
Plato
Re: Closed Expression for Series
Quote:

Originally Posted by veronicak5678
I thought that formula was for infinite series only? Can I use it if I go from 1 to n?

$\displaystyle \sum\limits_{k = 1}^n {r^k } = \frac{{r - r^{n + 1} }}{{1 - r}}$
• Aug 31st 2011, 10:55 AM
Prove It
Re: Closed Expression for Series
Quote:

Originally Posted by veronicak5678
I thought that formula was for infinite series only? Can I use it if I go from 1 to n?

Yes you are right. But there is a formula for finite geometric series as well, as Plato has explained.
• Aug 31st 2011, 11:07 AM
veronicak5678
Re: Closed Expression for Series
Using that formula for finite series is how I came up with the expression from my first post, 1 - 1/2^n. Is this correct? Is it closed?
• Aug 31st 2011, 11:10 AM
Prove It
Re: Closed Expression for Series
Quote:

Originally Posted by veronicak5678
Using that formula for finite series is how I came up with the expression from my first post, 1 - 1/2^n. Is this correct? Is it closed?

Yes it's correct. It's now closed because it is not a sum depending on k, it is a single value.
• Aug 31st 2011, 01:23 PM
veronicak5678
Re: Closed Expression for Series
OK. Thanks to all for helping.