Midterm 3 - Sequences and Series

UVA APMA 1110 Sequences and Series Review

Sequence

A list of numbers in a particular order. A sequence of numbers converges of lim a→n = L, where L is a finite number

Series

An infinite sum of a sequence. A series converges if it’s sequence of partial sums s_n converges to a finite number

Geometric Series

• A constant raised ot the power of n

• Converges if |r|<1, diverges otherwise

• Sum: If it converges, it adds up to S = a over 1-r

p-series

• The sum of 1 over n^p

• n raised to constant power in the denominator

• Converges if p>1, diverges of p<=1

Divergence test

• Take limit of terms as n → infinity

• If lim a_n does not = zero, series diverges

Alternating series test (AST)

Looks like (-1)^n-1 b_n, converges if:

1. The terms decrease in magnitude

2. The limit of the terms is zero

*AST only proves conergence, not divergence

The Ratio Test

• Best tool for factorials + mixed exponential/polynomial terms

• Used for power series

L = lim_n→Infinity | a_n+1 / a_n|

• If L < 1 → Converges

• If L > 1 → Diverges

• If L = 1 → Inconclusive

The Root Test

• Use when something is being raised to the n

L = lim_n→infinity of the nth root of |a_n|

• If L < 1 → Converges

• If L > 1 → Diverges

• If L = 1 → Inconclusive

Conclusions for ratio and root test (convergence or divergence)

• If L < 1 → Converges

• If L > 1 → Diverges

• If L = 1 → Inconclusive

Direct comparison test

Compare to simpler p-series or geometric series. All terms must be positive

Limit comparison test

If innequality in DCT goes the wrong way

lim_n→infinity a_n over b_n = C

If C>0 and finite, both series share the same fate

The integral test

If a_n = f(n) and f(x) is:

• Positive

• Continuous

• Decreasing

Then

The sum and integral eihter both converge or both diverge

Absolute Convergence

Apsolutely convergent: The sum of |a_n| converges

Conditionally Convergent

The series a_n converges but it’s apsolute value |a_n| diverges

Power series

• A series that has a variable c_n(x-a)ⁿ centered at a

• How to find the interval of convergence?

  • Use the ratio test and force the limit to be < 1

  • Solve the resulting inequality to find the radius of convergence (R)

  • Manually plyg endpoint values back into the original series and test using the convergence toolkit to see if they need [] or ()

Taylor and Maclaurin Series

• Allow you towrewrite common functions as infinite polynomaials

Taylor Series Formula:

Maclaurin Series Definition

A taylor series centered at a = 0

Taylor Series: e^x

e^x = xⁿ over n!

R = infinity

Taylor Series: cos(x)

cos(x) = (-1)ⁿ times x²ⁿ over (2n)!

R = infinity

Taylor Series: sin(x)

sin(x) = (-1)ⁿ times x²ⁿ⁺¹ over (2n+1)!

R = infinity

Taylor Series: 1/1-x

1/1-x = xⁿ

R = 1

Taylor Series: ln(1-x)

ln(1-x) = xⁿ⁺¹ over n+1

I = [-1,1]

Taylor Series: arctan(x)

arctan(x) = (-1)ⁿ times x²ⁿ⁺¹ over 2n+1

I = [-1, 1]

Estimating Error: Alternating Series Estimation

the error |E_n| is less than or equal to the magnitude of the very next term you left off (b_n+1)

Estimating Error: Taylor Polynomails

|R_n(x)| <= M|x-a|ⁿ⁺¹ over (n+1)! where M is the apsolute maximum of the next derivative, |fⁿ⁺¹(u)|, on the interval between your center a and estimate x