Alternating Series Estimation Theorem

1. Dec 27, 2004

Consider the following:

$$\int _0 ^1 \sqrt{1+x^4} \mbox{ } dx = \left[ x + \frac{x^5}{2\cdot 5} - \frac{1}{2!2^2 9}x^8 + \frac{1\cdot 3}{3!2^3 13}x^{12} - \frac{1\cdot 3\cdot 5}{4!2^4 17}x^{16} +\dotsb \right] _0 ^1$$

According to the alternating series estimation theorem, we find:

$$\left| R_n \right| \leq b_{n+1} < \left| \mbox{ error } \right| \Longrightarrow \frac{1\cdot 3}{3!2^3 13} < 10^{-2} \Longrightarrow \int _0 ^1 \sqrt{1+x^4} \mbox{ } dx \approx 1 + \frac{1}{2\cdot 5} - \frac{1}{2!2^2 9} \approx 1.09$$

The limits up there are easy to work with. So, how about if we'd had

$$\int _{0.7} ^{1.5} \sqrt{1+x^4} \mbox{ } dx$$

instead? Do we need to take into account those limits when applying the alternating series estimation theorem? I mean:

$$\left| R_n (x) \right| \leq b_{n+1} (x) < \left| \mbox{ error } \right|$$

Thanks​