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Suppose we have a power series $\sum a_n x^n$ with some positive radius of convergence whose coefficients are known. Let $f(x) = \sum a_n x^n$ within the radius of convergence. When truncating the sequence, how can we know when the partial sum is less than or greater than $f(x)$?

Here's a simple example to illustrate my question. If $f(x) = a_0 - a_1^2 x - a_2^2 x^2 - a_3^2 x^3 - \cdots$ then, if $S_n(x)$ denotes the $n^{\text{th}}$ partial sum of the power series, we have $f(x) \leq S_n(x)$ for all $n \geq 0$ and all $x$ within the radius of convergence.

But what if the coefficients are not all negative? When is it true that $S_n(x) \geq f(x)$ (or the reverse)? And when does the inequality hold outside of the radius of convergence?

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    To expand on Jacob's remark, for an _alternating_ series whose terms decrease in absolute value to zero, you always know the sign of the remainder to be the same as the sign of the first omitted term.2012-02-24

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