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
Is "n^√a" supposed to be or ? Assuming that it is the second, it is easy to see that . You can then expand using the binomial theorem to show that a= 1+ nx+ higher power terms so that a/n< x+ positive terms.