Theory

Theory 1

It can help to associate certain “strategy tips” to find convergence tests based on certain patterns.

Matching powers → Simple Divergence Test

$${\displaystyle \sum _{n=1}^{\infty }\frac{n-1}{2n+1}}$$

Use the SDT because we see the highest power is the same ($=1$) in numerator and denominator.

Rational or Algebraic → Limit Comparison Test

$$\sum _{n=1}^{\infty }\frac{\sqrt{n^3+1}}{3n^3+4n^2+2}$$

Use the LCT because we have a rational or algebraic function (positive terms).

Not rational, not factorials → Integral Test

$$\sum _{n=1}^{\infty }n{e}^{-n^2}$$

Use the IT because we do not have a rational/algebraic function, and we do not see factorials.

Rational, alternating → AST, and LCT or DCT

$$\sum _{n=1}^{\infty }{(-1)}^n\frac{n^2}{n^4+1}$$

Notice large $n$ behavior is like $n^2/n^4$ or $1/n^2$. This converges. Use the LCT to show absolute convergence. Skip the AST because absolute convergence settles the matter. Lesson: check for absolute BEFORE applying AST, even when alternating!

Factorials → Ratio Test

$$\sum _{n=1}^{\infty }\frac{2^n}{n!}$$

Use the RaT because we see a factorial. (In case of alternating + factorial, use RaT first.)

Recognize geometric → LCT or DCT

$$\sum _{n=1}^{\infty }\frac{1}{2+3^n}$$

Use the LCT or DCT comparing to $\frac{1}{3^n}$ because we see similarity to $\frac{1}{3^n}$ (recognize geometric).