Write down the taylor polynomial

Question

Problem:

Taylor's Theorem

1 Write down the Taylor polynomial P_n,0 (x) of 1n(x+1),x>-1. Use Taylor's theorem to

show that Σ^∞_n=1 (-1)n+1 1/n = 1 - 1/2 +1/3 - 1/4 + 1/5 + ... = 1n(2).

2. Given that (e^x)' = e^x, 1 < e < 3. Do the followings.

a) Write down the Taylor polynomial P_n,0 (x) of e^x and show that for and

       1+x/1! + ... + xⁿ/n! < e^x < 1+x/1!+...+xⁿ/n!+e^axⁿ⁺¹/(n+1)! for 0<x<a and n>1

b) Use a) to show that o>2 c) Conclude that e is irrational?