# Help needed on two questions!

• Dec 11th 2010, 10:05 AM
NFS1
Help needed on two questions!
Hi, i have recently been trying to complete an assignment of 10 mathematical questions. I have managed to answer 8 of the 10 questions but unfortunately have been unable to answer two. Can some please show me how to calculate the following as I have no idea how to attempt them. Below are the questions:

1) Given that the Maclaurin expansion of 1/1+x = 1 - x + x^2 - x^3... deduce the expansion of 1/1+cos(x) as far as the term in x^3.

2) If a>1 and n^√a = 1+x , prove that 0 < x < a/n. Deduce that n^√a --> 1 and n --> infiniti (∞). What is the corresponding result if 0 < a < 1?

Can anyone show me a step by step solution so i get a better understanding on how to answer these questions.

Thanks
• Dec 11th 2010, 10:24 AM
SammyS
Quote:

Originally Posted by NFS1
1) Given that the Maclaurin expansion of 1/(1+x) = 1 - x + x^2 - x^3... deduce the expansion of 1/(1+cos(x)) as far as the term in x^3.

So, $\displaystyle {{1}\over{1+\cos(x)}}=1-\cos(x)+\cos^2(x)-\cos^3(x)+\dots$

Do you know the Maclaurin expansion of $\displaystyle \cos(x)$ ?

Plug that in for each cosine in your expansion & do some algebra.
• Dec 12th 2010, 02:49 AM
HallsofIvy
Is "n^√a" supposed to be $n^{\sqrt{a}}$ or $\sqrt[n]{a}$? Assuming that it is the second, it is easy to see that $x= \sqrt[n]{a}- 1> 0$. You can then expand $(1+ x)^n= a$ using the binomial theorem to show that a= 1+ nx+ higher power terms so that a/n< x+ positive terms.