Why Does the k Term Change in Taylor Series?

[Physics2341313]

Have a quick question about taylor series. We covered taylor series somewhat in class, but there was a complete lack of explanation and our calculus book literally covers the topic in a single page.
I understand the idea of a taylor series and how its related to a power series, but what I don't understand and neither does anyone else is why the k term in a taylor series is sometimes a k+1 etc as in why it changes.
For example on some taylor series we will have (-1)^k but on others we will have (-1)^k+1 and for the factorial we will have k! or sometimes 2k! etc etc. Why does this happen? Just looking for someone to point me in the right direction or give an explanation.

Also, are there any good summarized notes available online for the series tests, have a midterm in a few days and we have barely covered any of this stuff, so I'm kind of lost at this point.

 

[Mark44]

Have a quick question about taylor series. We covered taylor series somewhat in class, but there was a complete lack of explanation and our calculus book literally covers the topic in a single page.

Wow! That's a pretty sparse treatment. Who's the author of your book?
I understand the idea of a taylor series and how its related to a power series, but what I don't understand and neither does anyone else is why the k term in a taylor series is sometimes a k+1 etc as in why it changes.
For example on some taylor series we will have (-1)^k but on others we will have (-1)^k+1 and for the factorial we will have k! or sometimes 2k! etc etc. Why does this happen? Just looking for someone to point me in the right direction or give an explanation.

The different values of k, k + 1, etc. are so that the expression in the summation matches the terms in the series. Some series start with k = 0 and others start with k = 1. You can start the series with an arbitrary value of k (an integer, though) by adjusting the exponents and subscripts in the summation formula.

Also, are there any good summarized notes available online for the series tests, have a midterm in a few days and we have barely covered any of this stuff, so I'm kind of lost at this point.

I would try wikipedia, searching for Taylor series and/or power series. They should probably have a reasonable summary.

 

[platetheduke]

You can think of Taylor series as polynomials with infinitely many terms. The way they are expressed in sigma notation is arbitrary and typically aimed at simplifying the indices, but there are many ways of indexing the same series. The standard way of writing common functions in sigma notation (like exponentials and trig functions) makes the notation simple -- but is not unique.

If a series has $(-1)^k$ then the terms with $k$ odd have negative signs, and $(-1)^{k+1}$ gives terms with $k$ even negative signs. It's really just a matter of what the series looks like expanded, and then how you want to condense it.