Integral Test

The integral test states that a series $\sum_{{n=1}}^{{\infty }}$an converges absolutely if these exists a positive decreasing function f such that f (n) = $\left\vert\vphantom{ a_{n}}\right.$an$\left.\vphantom{ a_{n}}\right\vert$ for each positive integer n and

$\displaystyle \int_{{1}}^{{\infty }}$f (x)dx < ∞


To verify convergence of $\sum_{{n=1}}^{{\infty }}$${\frac{{n^{2}}}{{2^{n}}}}$ using the integral test, define f by f (x) = ${\frac{{x^{2}}}{{2^{x}}}}$ and note the following.

$\blacktriangleright$ Evaluate, Evaluate Numerically

$\dint_{{1}}^{{\infty }}$${\dfrac{{x^{2}}}{{2^{x}}}}$dx = ${\dfrac{{1}}{{2}}}$${\dfrac{{%
2+2\ln 2+\ln ^{2}2}}{{\ln ^{3}2}}}$ = 5.805497209

Thus, this integral is finite. (Although for f (x) = ${\frac{{x^{2}}}{{2^{x}}}}$, it is true that f (1) < f (2) < f (3), you can verify that f is decreasing for x > 3. In fact,

f(x) = 2$\displaystyle {\frac{{x}}{{2^{x}}}}$ - $\displaystyle {\frac{{x^{2}}}{{2^{x}}}}$ln 2


is positive only on the interval 0 < x < ${\frac{{2}}{{\ln 2}}}$ = 2.885390082, so f is decreasing on 3 < x < ∞. Since convergence of a series depends on the tail end of the series only, it is sufficient that the sequence of terms be eventually decreasing.)