Does formula exist for this sum?

In summary, the given conversation discusses a series with a variable inside the sum and suggests using a geometric sum to solve it. The solution involves forming an auxiliary function and differentiating it to find the original sum. Wolfram Alpha can also be used to solve such problems.
  • #1
db453r
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0
[itex]\sum_{i=1}^{n}[i/2^i][/itex]

Have looked and looked and cannot find it anywhere.

EDITED: To correct mistake.
 
Last edited:
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  • #3
db453r said:
[itex]\sum_{i=1}^{n}[n/2^n][/itex]
That one's easy. There's nothing inside the sum that depends on i, so your sum is the same as ##\frac n {2^n}\sum_{i=1}^n 1##.

Do you mean ##\sum_{i=1}^n \frac i {2^i}## ?

Have looked and looked and cannot find it anywhere.
Have you tried Wolfram alpha, www.wolframalpha.com ?
 
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  • #4
Oops. Yeah, that's what I meant.
 
  • #5
Wow. Didn't know Wolfram could do that. Thanks.

Here's what it gave me:

[itex]\sum_{i=0}^{n} i/2^{-i} = 2^{-n}(-n+2^{n+1} -2)[/itex]
 
  • #6
You can solve this by hand by using a neat trick.
Form the auxiliary function (*):
[tex]F(x)=\sum_{i=1}^{i=n}(\frac{x}{2})^{i}[/tex], that is, F(x) is readily seen to be related to a geometric sum, with alternate expression (**):
[tex]F(x)=\frac{1-(\frac{x}{2})^{n+1}}{1-\frac{x}{2}}-1[/tex]
Now, the neat trick consists of differentiating (*), and we get:
[tex]F'(x)=\sum_{i=1}^{i=n}i*x^{i-1}2^{-i}[/tex]
that is, we have:
[tex]F'(1)=\sum_{i=1}^{i=n}i*2^{-i}[/tex]
which is your original sum!

Thus, you may calculate that sum by differentiating (**) instead, and evaluate the expression you get at x=1
:smile:
 

1. What is a formula for a sum?

A formula for a sum is a mathematical expression or equation that can be used to calculate the total value of a set of numbers being added together.

2. How do I find the formula for a sum?

To find the formula for a sum, you can use various methods such as using known mathematical patterns, applying algebraic equations, or using mathematical software or algorithms.

3. Is there a universal formula for all types of sums?

No, there is no one universal formula for all types of sums. Different types of sums, such as arithmetic, geometric, or infinite sums, may require different formulas or methods to calculate their values.

4. Can I create my own formula for a specific sum?

Yes, you can create your own formula for a specific sum by understanding the underlying mathematical concept and applying mathematical principles to derive an equation that accurately represents the sum.

5. Why is it important to have a formula for a sum?

A formula for a sum is important because it allows for quick and accurate calculation of the sum's value, without having to manually add up all the numbers. It also helps in understanding the patterns and relationships within the numbers being added together.

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