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## Homework Statement

Hi there,

I have just started taylor series for my course.. most seems arlgiht so far, however when it comes to validating a given series( tayor or maclaruin), I have an idea on how to find out the x value.. but I don't know what I am doing wrong.

Take for example:

**The following function (1+x)^-1 . Expand the maclaurin series of it and give values of x for which the above series is valid:**

## Homework Equations

F(x) = f(0) + f'(0)x /1! + f'' (0)x^2/ 2! + f'''(0)x^3 /3! + f^n (0)x^n/n!

## The Attempt at a Solution

Here's my working:

Let f(x) = (1+x)^-1 so f(0) = (1+0)^-1 = 1

f'(x) = -1. (1+x)^-2 f' (0) = -1 . (1)^-2

f''(x) = 2.1 (1+x)^-3 = > f''(0) = 2.1 (1)^-3

f'''(x) = -3.2.1. (1+x)^-4 => f'''(0) = -3.2.1 (1)^-4

Now initially i was puzzled on how to predict (next term or the general pattern) but I think next term will be f'4 = 4.3.2.1 (1x)^-5 so f'4 (0) = 4.3.2.1 (1)^-5

Now I know test ratio can be done to find the radius of convergence i.e for a given function say anx^n ratio would be => (an+1)x^(n+1) / anx^n....

I tried doing the same for the above series.. My general formula is : f(n) : (-1)^n . n! (1+x)^ -(n+1) .... by manipulation i got : 1.(n+1) x [ 1^-n-2] / 1^-n-1]

= > 1. (n+1) but since n => infinity , this function diverges ?

Any constructive input will be appreciated. Thanks!

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