Question about shifting the indexes [nth powers] and counting numbers of series?

In summary: HallsofIvy,Thanks for answering! As you can see, if you change the index on either series, the problem simplifies.
  • #1
nontradstuden
54
0
Question about shifting the indexes [nth powers] and counting numbers of series?
When you multiply a series by x, therefore changing the nth power on x, is it mandatory that you change the number where you start counting? This is confusing me because some problems do both, where others explain it in two different separate steps.

Basically,
If I am multiplying a taylor series by x, then am I supposed to add one to the index AND count one lower? or is it only mandatory to change the power on x, not the counting number?

If the index [nth power on x] is changed, is it mandatory to change where the series starts counting at?

I keep seeing problems where they only change one, and not the others. I thought you're supposed to replace all n's by the new relation.

I hope this makes sense, lol.
 
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  • #2
If your series is [itex]\sum_{n=0}^\infty a_nx^n[/itex] and you multiply by x, you will have [itex]\sum_{n=0}^\infty a_nx^{n+1}[/itex]. You can, although you don't have to let j= n+1. Then n= j- 1 so [itex]a_n= a_{j-1}[/itex] and the lower limit n= j-1= 0 become j=1. That is the series changes to [itex]\sum_{j=1}^\infty a_{j-1}x^j[/itex].
 
  • #3
@HallsofIvy,

Thanks for answering!

I have a couple more questions, if you will...

If you change the starting number of the series, then you don't have to change 'n' everywhere else? or not.

If you change the nth power on 'x' w/o multiplying by another 'x', then you must / don't have to change the starting number of the series?


I haven't had a problem with changing n everywhere before, but I'm working a problem where I'm not getting the right solution.

It's a problem where I'm multiplying a power series of (x-1)^(n-1) by x, so I changed x to [1+ (x-1)], but it isn't working out right.
 
  • #4
nontradstuden said:
...

I haven't had a problem with changing n everywhere before, but I'm working a problem where I'm not getting the right solution.

It's a problem where I'm multiplying a power series of (x-1)^(n-1) by x, so I changed x to [1+ (x-1)], but it isn't working out right.
So that gives you x(x-1)(n-1) = [1+ (x-1)](x-1)(n-1) = (x-1)(n-1) + (x-1)(n).

In this case it looks like shifting the index on either one of the series may simplify things.



.
 

1. What is meant by "shifting the indexes" in a series?

Shifting the indexes, also known as shifting the origin, refers to changing the starting point of a series. This changes the value of each index in the series, and thus the entire series can be shifted up or down.

2. How do you shift the indexes in a series?

To shift the indexes in a series, you need to add or subtract a constant value from each index. For example, if you want to shift the series up by 3, you would add 3 to each index. This will result in the first index being 3, the second index being 4, and so on.

3. Why would someone want to shift the indexes in a series?

Shifting the indexes in a series can be useful for various reasons. It can make the series easier to work with or align it with other data sets. It can also help in simplifying calculations or in visualizing patterns in the data.

4. Can shifting the indexes change the values in a series?

No, shifting the indexes does not change the values in a series. It only changes the starting point of the series and the value of each index, not the actual data in the series. The values in the series remain the same, but their corresponding indexes are different.

5. Is there a limit to how much you can shift the indexes in a series?

Technically, there is no limit to how much you can shift the indexes in a series. However, it is important to consider the range of values in the series and make sure that the shifted indexes do not exceed that range. Otherwise, it may result in errors or loss of data.

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