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32 Terms

1
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growth rates

(ln(n))p << nq<< an<< n!<<nn

2
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test for divergence

diverges: if lim as n—>∞ of an≠0, then ∑an diverges

This test cannot be used to show convergence

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Geometric series converges and diverges when?

Series: ∑arn

Converges: ∣r∣<1

Diverges: ∣r∣≥1

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Formula for the sum of geometric series

S: a1(first term)/1-r.

Σarn

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telescoping series

series: Σ(bn-bn+1)

converges= lim as n—>∞ of bn=L

diverges: cant be determined

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telescoping series sum formula

S=b1-L

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Generalized p-Test

series:∑1/np(ln(n))q

Converges: p>1, and if p=1 and q>1 when there is ln(n)

Diverges: p≤1, p=1 and q ≤1

Just like with the Integral Test the p-Series Test does not tell us
what a convergent series sums to.

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Alternating series

∑(-1)nbn

Converges: 0<bn+1≤bn, and lim as n—>∞ of bn=0

Diverges: cant be determined

an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit.

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Integral (f continuous, positive, and decreasing)

Series ∑an, an=f(n) ≥0

Converges: ∫ 1 to ∞ converges

Diverges: ∫ from 1 to ∞ diverges

Warning: The Integral Test only shows whether a series converges or diverges.
It does not tell us what a series converges to. In particular, the value of the
improper integral is not the sum of the series

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Root

converges: lim n—>∞ of √∣an∣<1

diverges: lim n—>∞ of √∣an∣>1

Test is inconclusive if √∣an∣=1

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Ratio

convergence: lim n—>∞ ∣an+1/an∣ < 1

divergence: lim n—>∞ ∣an+1/an∣ > 1

test is inconclusive if ∣an+1/an∣= 1

Notes:
-If the Ratio Test shows a series converges, then that series will converge
absolutely.
- If a series converges conditionally, then the Ratio Test will always be in-
conclusive.
-The Ratio Test is always inconclusive if an is a rational function. In this
case use the Limit Comparision Test.

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Direct comparison test

convergence: bn is > an, and bn converges, so both converge

divergence: bn is < an, and bn diverges, so both diverge

Warning: Both series must be series of positive terms in order to use the
Direct Comparison Test.


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limit comparison test

convergence: an/bn= L, and L is positive and bn converges, both converge

divergence: an/bn= L and L is positive and bn diverges, both diverge

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In general, if X is a continuous random variable, then the probability that
a ≤ X ≤ b is given by


P(a ≤ X ≤ b) =
ab f(x)

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to be a probability density function, what conditions must be satisfied:

1. f (x) ≥ 0 for all x, and
2. ∫∞-∞ f(x)=1

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The mean of a probability density function f (x) is given by

μ =∫∞-∞ x*f(x) dx

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The median is the number m such that


m∞ f(x) dx = 1/2

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work formula

W = F d
Wi = F (xi)∆x

W = ∫ab F(x) dx

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Alternating series test conditions:

There are three conditions that must be satisfied in order to use the Alternating Series Test:

1. bn ≥ 0 for all n (positive)
2. {bn} is decreasing
3. lim n→∞ bn = 0

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Absolute vs conditional convergence

A series is absolutely convergent if the series |an| converges.

If an converges but |an| diverges, then an is conditionally convergent

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Alternating Series Estimatation Theorem

: Let bn ≥ 0 for all n such
that {bn} is decreasing and l’m n→∞ bn = 0. Let (−1)nbn = S and let SN =
NX
n=0
(−1)nbn. Then |S − SN | ≤ bN +1

The Alternating Series Remainder Estimate says that the error obtained by us-
ing the partial sum SN to estimate the true sum of the convergent alternating
series ∑ (−1)nbn is never more than bN +1, the next term in the series.


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useful limits

1. lim
n→∞ (1 + a/n)bn= eab
2. lim
n→∞ n√|p(n)| = 1

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lim to infinity of bigger expo/smaller exp

lim= infinity

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lim to infinity of smaller expo/bigger expo

lim=0

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sequence is monotonic and bounded, then…

sequence is monotonic and bounded, then the sequence converges.

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(ln(n))p << nq<< an<< n!<<nn

something farther right/farther left

bigger/smaller

Infitinity

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(n))p << nq<< an<< n!<<nn

something farther left/farther right

smaller/bigger

0

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spring problem formulas

F=kx

1 cm= 0.01 m, 25/0.1=250

F=k(difference between natural and stretched length).

W=f*d

integrate using f=kx with found k value and the given stretch lengths

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work chain problem equations

w=f*d

work= mass*deltax*acceleration*xi

or….

top:

mass*deltax*acceleration*xi

bottom (chain left over):

work= mass*deltax*acceleration*distance

2*deltax*g*30(lower value of integration bounds)

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cubic tank questions

w=f*d

mass*acceleration*distance

mass=volume*density

volume=area*delta x

so…

integrate using the bounds where water is:

volume*deltax*density*g*xi

Area in cubic tank=area of bottom (4×4)

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cone problems

use simiar triangles to find radius to height

radi= 2, xi=4, so….. r=xi/2

w=f*d

volume*density*g*xi

Volume=Area*deltaX

area = pi(r)²*deltax= area = pi(xi/2)²*deltax

so………..

pi(xi/2)²*deltax*density*g*xi

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probabiliy density function conditions

1: positive for all x values

  1. integration from negative to pos. infinity =1

mean= neg in post inf= integration of x(fx)

median= itegation from m to infinirt= 1/2