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Fredy L.

Sequence

Ordered list of numbers {a_{1},a_{2},a_{3}...}

Recurrent Relation

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

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explicit formula

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

Infinite Series

a_{1}+a_{2}+a_{3}+.... = Sum from k=1 to infinity of a_{k}

Limit of a Sequence

limit as n approaches infinity = L exists, the limit converges to L. If the limit does not exist, then the sequence diverges

Sequence of Partial Sums

S_{1} = a_{1}

S_{2} = a_{1} + a_{2}

S_{3} = a_{1} + a_{2} + a_{3}

limit as n approaches infinity of (a_{n}+/-b_{n})

= A+/-B

limit as n approaches infinity of ca_{n}

= cA, where C is a real number

limit as n approaches infinity of a_{n}b_{n}

=AB

limit as n approaches infinity of a_{n}/b_{n}

= A/B, provided B does not equal 0

Geometric sequences:

Let r be a real number, then

limit as n approaches infinity of r^n= 0 if |r|<1,

1 if r=1,

DNE if r<-1 or r>1

If r>0, then {r^{n}} converges or diverges monotonically.

If r<0, then {r^{n}} converges or diverges by oscillation

Squeeze Theorem for Sequences

Let {a_{n}},{b_{n}},{c_{n}} be sequences with a_{n}<b_{n}<c_{n}.

If limit as n approaches infinity of a_{n} = limit as n approaches infinity of c_{n}, then limit as n approaches infinity of b_{n} = L.

To evaluate infinite series:

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

Geometric Series

Let a and r be real numbers. If |r| < 1, then sum from k=0 to infinity of ar^{k} = a/1-r.

If |r| > 1, the series diverges.

Divergence Test

If sum of ak converges, then lim as k approaches infinity of ak = 0. If limit does not equal 0, then series diverges.

**Cannot be used to determine convergence.

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Harmonic Series

Sum as k=1 to infinity of 1/k = 1 + 1/2 + 1/3 + 1/4 +....., diverges - even though terms of the series tend to 0

Integral Test

f is continuous, positive and increasing for x>1 and let a_{k} = f(k) for k=1,2,3... Then

Sum from k=1 to infinity of ak and integral from 1 to infnity of f(x)dx either both converge or both diverge.

*If both converge, the value of integral does not equal value of series.

Convergence of p-series

Sum as k=1 to infinity of 1/k^{p} converges when p>1 and diverges when p<(or equals) 1

Ratio Test

Let sum of a_{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

Root Test

Let sum of ak be an infinite series with nonnegative terms and let p = limit as k approaches infinity of ^{k}root(a_{k})

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

Comparison Test

Let sum of a_{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

Limit Comparison Test

Suppose that sum of a_{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

Alternating Series Test

Sum of (-1)^{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

Alternating Harmonic Series

Sum from k=1 to infinity of (-1)k+1/k = 1- 1/2 + 1/3 - 1/4 + 1/5...... converges although harmonic series diverges

Remainder in Alternating Series

R_{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}

Absolute and Conditional Convergence

Assume Sum of a_{k} converges.

converges absolutely if Sum of |a_{k}| converges. If not, then converges conditionally.

0 x infinity

Indeterminate

infinity - infinity

Indeterminate

K/infinity

0

infinity/K

infinity

0/infinity

0

infinity/0

infinity

O^{0}

Indeterminate

infinity^{0}

Indeterminate

K^{infinity}

k>1 then = infinity

0<k<1 then = 0

0^{infinity}

0

infinity^{infinity}

infinity

1^{infinity}

Indeterminate

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