Series Fundamentals

Question 1

When does series sum k=1 to inf Ak converge

Answer 1

Series sim k=1 to inf Ak converges when Sn = sum k=1 to n Ak converges to s as n tends to infinity.

Question 2

When is series sum k=1 to inf Ak absolutely convergent and what is conditional convergence

Answer 2

Series sum k=1 to in Ak is absolutely convergent when sum k=1 to inf mod(Ak) is a convergent series.
Conditional convergence is when a series is convergent but it absolutely convergent

Question 3

If a series is absolutely convergent, what else is it

Answer 3

If a series is absolutely convergent, then it is also convergent

Question 4

What is the vanishing test

Answer 4

Vanishing test is :
If series sum k=1 to inf Ak converges then lim k tends to inf Ak = 0

Question 5

What is the comparison test

Answer 5

Comparison test is:
Suppose sum k=1 to inf Ak and sum k=1 to inf are 2 series satisfying 0 <= Ak <= Bk for all k in N, if sum k=1 to inf Bk converges, then sum k=1 to inf Ak converges

Question 6

What is the ratio test

Answer 6

Ratio test is :
Series sum k=1 to inf Ak converges absolutely if 0 <= Lim k tends to inf supMod(Ak+1 /Ak) < 1 and diverges if 1 < lim k tends to inf inf mod(Ak+1 /Ak) <= inf

Question 7

What is the root test

Answer 7

Root test is :
Let Ak be a sequence of real numbers, gamma = lim k tends to inf sup mod(Ak)^1/k then series sum k=1 to inf Ak converges absolutely if 0 </ gamma < 1 and diverges of gamma > 1