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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    $\begingroup$ Except in very specific cases, it think it's hard to say something. For instance take the sine function. $S_n$ is a polynomial wich goes to $+$ or $- \infty$ depending on the parity of $n$. $\endgroup$ – Jacob Ikabruob Feb 24 '12 at 9:06
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    $\begingroup$ 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. $\endgroup$ – Harald Hanche-Olsen Feb 24 '12 at 10:33

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