How to solve summation equations with different indices

In summary: Combine pairs of terms.The terms for k being even are: ##\dfrac 1 {2j-1}## . There are n of these. You can write a general term for these as ##\dfrac 1 {2j}## , where j takes on values from 1 to n. Right?
  • #1
musicgold
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19
Homework Statement
I need to prove the equation below. What scares me is the fact that the LHS and RHS have different summation indices.
Relevant Equations
See below
1569938319222.png


To analyze the LHS of this equation, I used (k-1) , k and (K+1) to get

## \frac {(-1)^{k-1} } { (k-1)} \ . \frac {(-1)^k} { (k)} \ . \frac {(-1)^{k+1} } { (k+1)} \ ##

Nothing cancels out in these terms and the sign of each term is the opposite of the previous term.

I calculated the LHS for a k = 1, 2, 3, 4, 5, 6 . The sum is = 1 -1/2 + 1/3 -1/4 + 1/5 -1/6. It looks like an oscillating sum that is going towards 2/3.

How can I proceed from here to get to the RHS?

Thanks
 
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  • #2
Hi,

musicgold said:
the fact that the LHS and RHS have different summation indices.
they don't: on both sides ##k## is the summation index.

To analyze the LHS of this equation, I used (k-1) , k and (K+1) to get

## \frac {(-1)^{k-1} } { (k-1)} \ . \frac {(-1)^k} { (k)} \ . \frac {(-1)^{k+1} } { (k+1)} \ ##

Nothing cancels out in these terms
I don't see what you are doing here ...

musicgold said:
I calculated the LHS for a k = 1, 2, 3, 4, 5, 6 . The sum is = 1 -1/2 + 1/3 -1/4 + 1/5 -1/6
So here you wrote out (but did not add up) the LHS for ##n=3##

Now do the same for the RHS -- I get 37/60

Finally, recognize the pattern and see that an increase in ##n## increases both sides by the same amount.
 
  • #3
BvU said:
...

Now do the same for the RHS -- I get 37/60

Finally, recognize the pattern and see that an increase in ##n## increases both sides by the same amount.

I am trying to get to the RHS part of the original equation. I don't know how to change the summation index. I think it involves removing half the terms.
 
  • #4
There is no need to change anything. ##n## is the 'independent variable': for each value of ##n##, het LHS gives a value for the sum, and so does the RHS.

So: for ##n=3## you have an LHS value, now fill in ##n=3## on the RHS and evaluate the sum
 
  • #5
musicgold said:
I am trying to get to the RHS part of the original equation. I don't know how to change the summation index. I think it involves removing half the terms.

Precisely. It's not that there are "different summation indices". It's that there are twice as many terms on the left as on the right.

There is one term on the right corresponding to every two terms on the left.

Does that give you a hint what you might do?

I don't know what you mean by using k-1, k and k + 1. I'd suggest instead starting with n = 1. That gives you two terms on the left, one on the right. Write out all the terms explicitly. See if you can make that work.

Then try n = 2. That gives you four terms on the left but you already know the sum of the first two. Write out all the terms again, without summation notation.
 
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  • #6
BvU said:
There is no need to change anything. ##n## is the 'independent variable': for each value of ##n##, het LHS gives a value for the sum, and so does the RHS.

So: for ##n=3## you have an LHS value, now fill in ##n=3## on the RHS and evaluate the sum
I think I did not explain my question enough. Let's assume we don't know the RHS part and we are trying to simplify the LHS.

Here is what I am getting when I calculate values for different k and n values. How can I move ahead from here?

1569964586630.png
RPinPA said:
I don't know what you mean by using k-1, k and k + 1. I'd suggest instead starting with n = 1. That gives you two terms on the left, one on the right. Write out all the terms explicitly. See if you can make that work.

I am using k-1, k, and k+1 in the LHS part to see if there is a pattern.
 
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  • #7
musicgold said:
I think I did not explain my question enough. Let's assume we don't know the RHS part and we are trying to simplify the LHS.

Here is what I am getting when I calculate values for different k and n values. How can I move ahead from here?

1569964586630-png.png


I am using k-1, k, and k+1 in the LHS part to see if there is a pattern.
Combine pairs of terms.

The terms for k being odd are: ##\dfrac 1 {1}, \dfrac 1 {3}, \dfrac 1 {5}, \dfrac 1 {7}, \dots , \dfrac 1 {2n-1}## . There are n of these. You can write a general term for these as ##\dfrac 1 {2j-1}## , where j takes on values from 1 to n. Right?

The terms for k being even are: ##-\dfrac 1 {2}, -\dfrac 1 {4}, -\dfrac 1 {6}, -\dfrac 1 {8}, \dots , - \dfrac 1 {2n}## . There are also n of these. Can you you come up with an expression for a general term for these using an index (Why not also use j ?) which takes on values from 1 to n ?

Added in Edit:
You should be able to include the two expressions in a single summation, then use algebra to combine the two into a single expression.

By the way: The this post only addresses the LHS .
 
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  • #8
I worked with n=5 and n=6. I manually obtained the terms and summed to show the expression true for n=5. For n=6 , we will add (1/11 - 1/12) = 1/132 for the expression on the left. For the expression on the right, we will add two terms (1/11 + 1/12) but will loose one - ie 1/(5+1). Then 1/11+1/12-1/6=1/132 also.

You could do something similar with n=m and n=m+1 and hence prove by induction.
 
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  • #9
musicgold said:
Here is what I am getting when I calculate values for different k and n values. How can I move ahead from here?

1569964586630-png.png
To make discussion less clumsy, let's define ##L_k ## to be ##\dfrac{(-1)^{(k-1)}}{k} ## .

There is a fundamental problem with your table. You don't sum ##L_k ## from k = 1 to k = n. You sum it from k=1 to k = 2n. The only meaningful entries in the bottom row are for even values of k.

For n = 2, the sum is 7/12.

For n = 3, the sum is 37/60.

Notice that the difference between these is 1/30.
 
  • #10
musicgold said:
I am using k-1, k, and k+1 in the LHS part to see if there is a pattern
You don't want that. For a given value of ##n## LHS sums up to something, so does RHS. ##k## has nothing to do with it.

If you now increase ##n## by 1 to ##n+1## , LHS(n+1) = LHS(n) + ##{1\over 2n+1} - {1\over 2n+2}##

and unfortunately RHS(n+1) ##\ne## RHS(n) + ##{1\over (n+1) + (n+1)} ## but, as @neilparker62 indicated
RHS(n+1) = RHS(n) + ##{1\over 2n+1 } + {1\over 2n+2 } - {1\over n+1 }##

If we want the increments to be the same, we must show that $$
{1\over 2n+1} - {1\over 2n+2} = {1\over 2n+1 } + {1\over 2n+2 } - {1\over n+1 }$$which is not so hard ... 😉
 
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Related to How to solve summation equations with different indices

What is a summation equation?

A summation equation is a mathematical expression that represents the sum of a series of terms. It is denoted by the symbol Σ (sigma) and includes an index, a starting value, and an ending value.

What are indices in summation equations?

Indices in summation equations are the variable that changes with each term in the series. It is usually represented by a letter, such as n, k, or i, and helps to keep track of the different terms being added.

How do I solve summation equations with different indices?

To solve summation equations with different indices, you need to find a common index that all the terms share. You can do this by rewriting the indices in terms of a common index or by changing the indices to match the starting and ending values of the sum.

What is the difference between the upper and lower limits in summation equations?

The upper and lower limits in summation equations represent the starting and ending values of the sum. The upper limit is the last value that the index takes on, while the lower limit is the first value. The difference between the two is the number of terms in the series.

Can I use different indices in the same summation equation?

Yes, you can use different indices in the same summation equation as long as they are consistent with the terms being added. This means that the indices should follow a pattern and the corresponding terms should have the same index.

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