# Taylor Series - Range of values

## Homework Statement

im being asked for the first 4 non zero values for the taylor expansion of exp(x) which is simple, but then it asks for the range of x values that are valid for the expansion.

i have never come accross ths before - any idea?

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From the first four coefficients of the taylor expansion you can guess what the nth coefficient is going to be. Is there then some test you can use to find out when a sum (don't worry about it being a taylor series!) converges?

ratio test? integral test?

but how will working out if the sum convereges help with obtaining the valid x values?

The ratio test is useful for calculating when taylor series converge. Each of the terms of your sum is in terms of 'x' so the ratio test tells you when the sum converges in terms of x.

the question doesnt mention sum or convergence?

The taylor series expansion is going to be valid where the sum converges, or alternatively you can just write that the taylor series expansion for exp is valid everywhere if you've been told this fact.