Integral Test

#Calc-2#Math#Series If we treat the partial sums from a series as points on a graph,

Ignoring the first rectangle, we can write it as an integral.

$\sum _{k=1}^{\infty }\frac{1}{k^2}\le 1+{\int }_1^{\infty }\frac{1}{x^2}dx$ (the 1 accounts for the first term/rectangle)

We could also use a different Riemann Sum

If we combine these two we end up with,

${\int }_1^{\infty }\frac{1}{x^2}dx\le \sum _{k=1}^{\infty }\frac{1}{k^2}\le 1+{\int }_1^{\infty }\frac{1}{x^2}dx$ If the improper integral converges then the series is between two finite #‘s, so the series must also converge. Otherwise the series diverges.

Examples §