Estimating Error of Infinite Series by Finite Series

Question

My book gives the following explanation for finding the error ($ R_{10} $) associated with the sum of the first 10 terms of the following infinite series: $$ (1) \; R_{10}=\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^4+1}}-\sum_{n=1}^{10} \frac{1}{\sqrt{n^4+1}} $$ Since $$ (2) \;\frac{1}{\sqrt{n^4+1}} \le \frac{1}{n^2} $$ for all $ n $, then $ R_{10} \le T_{10} $, where $$ (3) \;T_{10}=\sum_{n=1}^{\infty} \frac{1}{n^2}-\sum_{n=1}^{10} \frac{1}{n^2} $$ It then goes on to find the error in $ T_{10} $ by taking the integral. What I don't understand is how $ (2) $ explains $ R_{10} \le T_{10} $. The first term of $(3)$ is greater than the second term of $(1)$, but it is also larger in the second term. In my mind, the weight of each term will determine whether $R_{10}$ or $T_{10}$ is larger. Am I looking at this in the wrong way?

Answer 1

Hint: If $$\dfrac{1}{\sqrt{n^4+1}}\leq\dfrac{1}{n^2}$$Then $$\sum^{n}_{n=1}\dfrac{1}{\sqrt{n^4+1}}\leq\sum^{n}_{n=1}\dfrac{1}{n^2}$$ That is very easy to show.