Why is Adding One Necessary in Binomial Series for 1/\sqrt{1-x^{2}}?

[paridiso]

When using a binomial series to expand $$1/\sqrt{1-x^{2}}$$ I come up with the correct answer except that I do not add the number one to my answer. Why do I have to add one to the series, should this not arise when calculating the sum?

 

[Dick]

If x is zero the result is 1. You should definitely have a 1 in the series. What DID you do?

 

[paridiso]

When expanding it I come up with the following result:

$$\sum (1*3* \ldots (2n-1)*x^{2n}) / 2^{n}n!$$

According to the answer key the answer is:

$$1 + \sum ((1*3* \ldots (2n-1)*x^{2n}) / 2^{n}n!)$$

Where does the one come from?

 

[Dick]

You didn't put limits on your summations. It looks like in the books answer the limits are 1 to infinity. In your answer they are 0 to infinity. Isn't 1 the n=0 term in your series?

 

[g_edgar]

Yes ... they separated out the $n=0$ term, because they thought the student would not understand the product [tex]1\cdot3\cdots(-1)[/itex][/tex]

 

[paridiso]

Thanks, I understand it now.