1. Approximating an infinite series

Approximate a sum for the infinite series:
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^4}$
with a maximum error of $\displaystyle 0.02$

$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^4}\approx 1.0748$

My attempt:
For every $\displaystyle N\in\mathbb{N}$:
$\displaystyle \int\limits_{N+1}^t\frac{1}{x^4}\, dx$$\displaystyle =\left [ \frac{1}{3x^3} \right ]_{N+1}^t=\frac{1}{3(N+1)^3}-\frac{1}{3t^3} For \displaystyle t\rightarrow \infty: \displaystyle \int\limits_{N+1}^t\frac{1}{x^4}\, dx$$\displaystyle =\frac{1}{3(N+1)^3}$

$\displaystyle f(x)=\frac{1}{x^4}$
$\displaystyle \int\limits_{N+1}^{\infty}f(x)\, dx$$\displaystyle =\int\limits_{N+1}^{\infty}\frac{1}{x^4}\, dx$$\displaystyle =\frac{1}{3(N+1)^3}$
$\displaystyle f(N+1)=\frac{1}{(N+1)^4}$

$\displaystyle \int\limits_{N+1}^{\infty}f(x)\, dx + f(N+1)=\frac{1}{3(N+1)^3}+\frac{1}{(N+1)^4}=\frac{ 1}{3}\frac{N+4}{(N+1)^4}$

Expanding we get:
$\displaystyle \int\limits_{N+1}^{\infty}f(x)\, dx + f(N+1)=\frac{1}{3}\frac{N+4}{n^4+4N^3+6N^2+4N+1}$$\displaystyle =\frac{N+4}{N(3N^3+12N^2+18N+12)+3} Since: \displaystyle N(3N^3+12N^2+18N+12)+3\ge N(3N^3+12N^2+18N+12) We get that: \displaystyle \int\limits_{N+1}^{\infty}f(x)\, dx + f(N+1)$$\displaystyle \le \frac{N+4}{N(3N^3+12N^2+18N+12)}\le \frac{1}{N(3N^3+12N^2+18+\frac{1}{3})}$

Which shows us that:
$\displaystyle \frac{1}{N(3N^3+12N^2+18+\frac{1}{3})}\le 0.02$

However when I try to solve for $\displaystyle N$ I get a negative $\displaystyle N$-value which doesn't make sense. Can somebody help me get the correct $\displaystyle N$

2. Re: Approximating an infinite series

Originally Posted by MathIsOhSoHard
Approximate a sum for the infinite series:
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^4}$
with a maximum error of $\displaystyle 0.02$

$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^4}\approx 1.0748$

My attempt:
For every $\displaystyle N\in\mathbb{N}$:
$\displaystyle \int\limits_{N+1}^t\frac{1}{x^4}\, dx$$\displaystyle =\left [ \frac{1}{3x^3} \right ]_{N+1}^t=\frac{1}{3(N+1)^3}-\frac{1}{3t^3} For \displaystyle t\rightarrow \infty: \displaystyle \int\limits_{N+1}^t\frac{1}{x^4}\, dx$$\displaystyle =\frac{1}{3(N+1)^3}$

$\displaystyle f(x)=\frac{1}{x^4}$
$\displaystyle \int\limits_{N+1}^{\infty}f(x)\, dx$$\displaystyle =\int\limits_{N+1}^{\infty}\frac{1}{x^4}\, dx$$\displaystyle =\frac{1}{3(N+1)^3}$
$\displaystyle f(N+1)=\frac{1}{(N+1)^4}$

$\displaystyle \int\limits_{N+1}^{\infty}f(x)\, dx + f(N+1)=\frac{1}{3(N+1)^3}+\frac{1}{(N+1)^4}=\frac{ 1}{3}\frac{N+4}{(N+1)^4}$

Expanding we get:
$\displaystyle \int\limits_{N+1}^{\infty}f(x)\, dx + f(N+1)=\frac{1}{3}\frac{N+4}{n^4+4N^3+6N^2+4N+1}$$\displaystyle =\frac{N+4}{N(3N^3+12N^2+18N+12)+3} Since: \displaystyle N(3N^3+12N^2+18N+12)+3\ge N(3N^3+12N^2+18N+12) We get that: \displaystyle \int\limits_{N+1}^{\infty}f(x)\, dx + f(N+1)$$\displaystyle \le \frac{N+4}{N(3N^3+12N^2+18N+12)}\le \frac{1}{N(3N^3+12N^2+18+\frac{1}{3})}$

Which shows us that:
$\displaystyle \frac{1}{N(3N^3+12N^2+18+\frac{1}{3})}\le 0.02$

However when I try to solve for $\displaystyle N$ I get a negative $\displaystyle N$-value which doesn't make sense. Can somebody help me get the correct $\displaystyle N$
Solving that 4th order equation for N is a nightmare. You can get an approximate value

$\displaystyle 3N^4< N(3N^3+12N^2+18+\frac{1}{3}) \iff \frac{1}{N(3N^3+12N^2+18+\frac{1}{3})} < \frac{1}{3N^4}$

This much better to solve

$\displaystyle \frac{1}{3N^4} < \frac{2}{100} \iff 100 < 6n^4$

You can see that $\displaystyle n=3$ works

3. Re: Approximating an infinite series

$\displaystyle \sum _{k=1}^{\infty } \frac{1}{k^4}$=$\displaystyle 1+\frac{1}{2^4}+\frac{1}{3^4}$

$\displaystyle \frac{1}{3^4}<.02$

4. Re: Approximating an infinite series

Hey MathIsOhSoHard.

Can you show us how you solved the roots of the equation N(3N^3 + 12N^2) + 18 + 1/3)*0.02 - 1 = 0? (Knowing the roots means you know which values of N will be on either side of the inequality).