opening by turki alqahtani

A list of numbers following an order

turkiq07 — 2021-04-05 00:29:28 UTC

Can be both infinite and finite

turkiq07 — 2021-04-05 00:29:44 UTC

A_n

turkiq07 — 2021-04-05 00:29:56 UTC

Sigma notation

turkiq07 — 2021-04-05 00:30:13 UTC

List of numbers added up all together

turkiq07 — 2021-04-05 00:30:24 UTC

tells convergence or divergence of an improper integral

turkiq07 — 2021-04-05 00:30:59 UTC

x^n: if n is greater than or equal to 1, the function converges to a constant value.
x^n: if n is less than 1, the function diverges.

F(x)

turkiq07 — 2021-04-05 00:31:46 UTC

Change series into a f(x) function and take the integral like an improper integral.

Integral test conditions

turkiq07 — 2021-04-05 00:32:03 UTC

The functions must be:
continuous
positive
Decreasing

Two conditions must be met:

turkiq07 — 2021-04-05 00:32:44 UTC

0 < An < Bn
If the series Bn converges then series An converges
If the series An diverges then series Bn diverges

turkiq07 — 2021-04-05 00:33:54 UTC

limit as n goes to infinity (a[n]/b[n])

turkiq07 — 2021-04-05 00:34:12 UTC

Lim as n goes to infinity An / Bn = L
If the series An converges then series Bn converges
If sereis An diverges then series Bn diverges
L=Positive finite value

Test for decreasing/limit

2021-04-07 21:03:52 UTC

Check if b(n) is decreasing and whether the limit of b(n) goes to zero. If both conditions are met, the series converges

lim n goes to infinity | An + 1/ An|

2021-04-07 21:04:18 UTC

| An + 1 / An | < 1

2021-04-07 21:04:48 UTC

converges

use the Ratio Test

2021-04-07 21:04:59 UTC

test with absolute values to tell if converge or diverge.

rules

2021-04-07 21:05:01 UTC

Absolute and Conditional Convergence

2021-04-07 21:05:19 UTC

If ratio test diverges, means conditional. Also need to test the alternate series test to see if the series is conditionally convergent. If the ratio test gives convergence, then absolute convergence.

|An + 1 /An| > 1

2021-04-07 21:05:58 UTC

diverges

|An + 1/ An| = 1

2021-04-07 21:06:54 UTC

inconclusive

Check Bn

2021-04-07 21:07:41 UTC

Based on Bn's behavior, Bn's behavior would be the same as An if there is a positive finite result from the limit.

Divergence

2021-04-07 21:10:17 UTC