Question 3

Question 1

Convergence:

Answer 1

an converges to a limit aeR if for all E>0 there exists NeN s.t. if n>N then /an-a/<E.

Question 2

Divergence:

Answer 2

if an is not convergent. divergence to +- infinity if for all M>0 there exists NeN s.t. an>M for +infinity or an<-M for -infinity.

Question 3

Absolute Convergence:

Answer 3

Sum of an from 1 to infinity is absolutely convergent if the sum of /an/ from 1 to infinity is convergent. Absolute convergence is a stronger property than convergence.

Question 4

Conditionally convergence:

Answer 4

Convergent but not absolutely convergent.

Question 5

Tests for radius of convergence:

Answer 5

Root test, ratio test, comparison test

Question 6

Root test:

Answer 6

Let an>0 for all neN and let an^1/n tend to l as n tends to infinity. if l>1 or l=infinity then sum of an from 1 to infinity diverges.
If l<1 the series of sum of an from 1 to infinity converges. If l=1 the test is inconclusive.

Question 7

Ratio test:

Answer 7

Let an>0 for all neN and let (an+1)/an tend to l as n tends to infinity. if l>1 or l=infinity then sum of an from 1 to infinity diverges.
If l<1 the series of sum of an from 1 to infinity converges. If l=1 the test is inconclusive.

Question 8

Comparison test:

Answer 8

Let an and bn be two sequences s.t. 0</an</bn for all neN.
i) if the sum of bn from 1 to infinity converges then the sum of an from 1 to infinity converges.
ii) if the sum of an from 1 to infinity diverges then the sum of bn from 1 to infinity diverges.

Question 9

Alternating series test:

Answer 9

If an is a decreasing non-negative sequence, an tends to 0 and n tends to infinity then the sum of (-1)^n*an from 1 to infinity is convergent, other than if an=1/n (Harmonic Series).

Question 10

Shift rule for series:

Answer 10

Let NeN, then the sum of an from 1 to infinity is convergent iff the sum of aN+n from 1 to infinity converges. Same applies if an is divergent.