MATH 1552 Exam 3 - types of series

Geometric Series

• form

• conv/div conditions

• sum

• a₁ is the first term in the series

• r is the ratio

P-series

• form

• conv/div conditions

Harmonic Series

• form

• conv/div conditions

• always diverges

• p-series where p = 1

Telescoping Series

• form

• conv/div conditions

• sum

to solve:

• plug in values and find the general formula

• take the limit of the general formula

• if the limit is a finite value, the series converges. Otherwise, the series diverges

• A result of a finite number means the series is “solved”

Alternating Series

• form

• conv/div conditions

• If the absolute value of the sequence is decreasing and the limit of the absolute value of the sequence = 0, the sequence converges

• check if the sequence decreases by comparison inequality or taking the derivative. Can also maybe plug in terms.

Steps:

1. divergence test of absolute value of the sequence

2. check decreasing

3. find type of convergence by looking at the absolute value of the series. If the series diverges, the original series converges conditionally. If the series converges, the original series converges absolutely. Can use any test

Power Series

• form

• conv/div conditions

• I.O.C. and R.O.C.

• has 2 “variables”

• x is treated like a number, and conv/div is dependent on x

To solve:

1. use Ratio/Root test to determine for which values x the series conv/div

2. after obtaining the inequality, solve for the “edge cases” of the series

3. plug in each “edge x” into the series and separately determine if each series conv/div

4. use all the info to determine I.O.C. and R.O.C

I.O.C. and R.O.C:

• I.O.C. = range of values for which the series converges

• R.O.C = half the length of the I.O.C.

  • If I.O.C. = all real numbers, R.O.C = infinity

  • If I.O.C. = just the center, R.O.C = 0