# Calc 2

**Created:**2014-05-10

**Last Modified:**2016-03-08

_{1},a

_{2},a

_{3}...}

_{n+1}=f(a

_{n}) for n=1,2,3...

_{n}=f(n) for n=1,2,3...

_{1}+a

_{2}+a

_{3}+.... = Sum from k=1 to infinity of a

_{k}

_{1}= a

_{1}

_{2}= a

_{1}+ a

_{2}

_{3}= a

_{1}+ a

_{2}+ a

_{3}

_{n}+/-b

_{n})

_{n}

_{n}b

_{n}

_{n}/b

_{n}

^{n}} converges or diverges monotonically.

^{n}} converges or diverges by oscillation

_{n}},{b

_{n}},{c

_{n}} be sequences with a

_{n}<b

_{n}<c

_{n}.

_{n}= limit as n approaches infinity of c

_{n}, then limit as n approaches infinity of b

_{n}= L.

- Determine a formula for sequence for partial sums
- Find its limit

^{k}= a/1-r.

_{k}= f(k) for k=1,2,3... Then

^{p}converges when p>1 and diverges when p<(or equals) 1

_{k}be an infinite series with positive terms and let r = limit as k approaches infinity of a

_{k+1}/a

_{k}

- If 0 <(or equals) r < 1, series converges
- If r >1 (infinity), series diverges
- If r = 1, test is inconclusive

^{k}root(a

_{k})

- If 0 <(or equals) p < 1, series converges
- If p>1(including infinity), series diverges
- p=1, test is inconclusive

_{k}and sum of b

_{k}be series with positive terms

- If 0 < a
_{k}< b_{k}and sum of b_{k}converges, then sum of a_{k}converges - If 0 < b
_{k}< a_{k}and sum of b_{k}diverges, then sum of a_{k}diverges

_{k}and sum of b

_{k}have positive terms and limit as k approaches infinity a

_{k}/b

_{k}= L

- 0 < L < infinity, then sum of a
_{k}and sum of b_{k}both converge or diverge - If L=0 and sum of b
_{k}converges then sum of a_{k}converges - If L = infinity and sum of b
_{k}diverges, then sum of a_{k}diverges

^{k+1}a

_{k}converges provided:

- Terms are nonincreasing in magnitude (0<a
_{k+1}<a_{k}) - limit as k->inf of a
_{k}= 0

_{n}= |S-S

_{n}|, remainder in convergent alternating series sum of k=1 to infinity of a

_{k}. Then R

_{n}<(or equal to) a

_{n+1}

_{k}converges.

_{k}| converges. If not, then converges conditionally.

^{0}

^{0}

^{infinity}

^{infinity}

^{infinity}

^{infinity}

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