Series

Question 1

Geometric Series

Answer 1

E ar ^n
Sum = a / (1-r)
Converges: |r|<1 (limit = 0)
Diverges: |r|>=1 (limit approaches infinity)
MUST INCLUDE |r| < 1!

Question 2

*Geometric Series (r, ar, when n = 1)

Answer 2

• whatever’s raised to the Nth degree is r (when series starts at 0)
• ar = 2nd term
• r = 2nd term / 1st term
  a = …, ar = .., ar = ar (divide to get r)

Question 3

Reindexing a Geometric Series

• If given summation of starting at n = 1, E #^n+1 / #^n-1 …
• If given summation of starting at n = 1, E (-1)^n-1 # / #^n

Answer 3

• so the true value of ‘a’ can be seen

Raise: start @n = 0, E #^(n+1)+1 / #^(n+1)-1
[ simplify, state a,r, |r| > 1 ]
n = # + 1, subtract 1 from starting n

Raise: start at n = 0, E (1-)^(n+1)-1 #/ #^(n+1)
[ simplify, state a,r, |r| > 1 ]
n = # + 1, subtract 1 from starting n

• Lower:
  n = 1 - h, add h to an

Question 4

Nth Term Test DIVERGENCE

Answer 4

if lim of an (n approaches infinity) DOES NOT EQUAL ZERO, it diverges

*lim = 0 means NOTHING (inconclusive)

Question 5

*Telescoping Series Method

Answer 5

• has both a positive and negative term
• Sk = the series at (n=1) + (n=2) + (n=3) +…
  … + (n=k-2) + (n=k-1) + (n=k)
• state sk after canceling out first and last
• take the limit
• MAY HAVE TO USE PARTIAL FRACTIONS

Question 6

Integral Test

Answer 6

• compare to
• integral from n to infinity
• replace n’s with x’s (dx)
• check positive ( > 0), decreasing (f’(x) < 0 OR an > an+1), solve integral (infinity = diverges, finite = converges)

Question 7

P-Series

Answer 7

• Summation of 1 / n ^p
• p = 1, harmonic series (1/n DIVERGENT)
• p > 1, converges p < 1, series diverges

Question 8

Direct Comparison Test

Answer 8

• given an choose bn, COMPARE LEADING TERMS
• state what bn does and how you know
• if bn converges, prove an < bn
• if bn diverges, prove an > bn
  anything else = inconclusive

Question 9

Limit comparison

Answer 9

• given an choose bn, COMPARE LEADING TERMS
• prove positive
• lim (n-> infinity) an / bn = # > 0, both converge or diverge
• lim (n-> infinity) an / bn = 0, AND bn CONVERGE, then an converges
• lim (n-> infinity) an / bn -> infinity, AND bn DIVERGE, then and diverges
• else: lim (n->infinity) bn / an = conclusive answer

Question 10

Alternating Series Test (AST)

Answer 10

(-1)^n or (-1)^n+1

• must exist to use AST, but is IGNORED when testing
• positive, decreasing (an > an+1), lim as n -> infinity = 0

Question 11

Absolute Convergence

Answer 11

• compare to series with abs values, if converges, converges
• root / ratio test if lim < 1 converges absolutely
• power series, get limit into |x-a| < r form
• -1 < |x-a| < 1 int of abs conv

Question 12

Ratio Test

Answer 12

• must prove positive or use abs values
• lim of an+1 / an as n -> infinity
• < 1, converges absolutely
• > 1 or -> infinity, diverges
• = 1, INCONCLUSIVE (usually with p-series)

Question 13

Root Test

Answer 13

• must prove + or use abs values
• lim of the nth root of an as n -> infinity
• < 1, converges absoulutely
• > 1 or -> infinity, diverges
• =1, INCONCLUSIVE (usually with p-series)

Question 14

Power Series

Answer 14

• has a x in it

Question 15

Testing Power Series for Absolute Convergence

Answer 15

• use Root / Ratio test to find interval of absolute convergence
• |x-a| < R

Question 16

Find the power series representation and radius of convergence

Answer 16

f(x) = integral E
-> f’(x) -> E ->

    ->  integral f(x)dx  ->  E ->  f(x)                   =                  d/dx E

get into i / a - r form, write converges for |r|<1, simplify to get R

Question 17

Interval of Convergence

Answer 17

• solve limit
• | | < 1
• put x term in the middle of -1 1 inequality and simplify
• -> interval of absolute convergence
• test endpoints (conditional convergence) together = int. of conv.

Question 18

Taylor Polynomial vs Maclaurin + formula

Answer 18

Taylor a = given, Maclaurin a = 0 always

• derivatives, evaluate
• f(a) + f’(a)(1/1!)(x-a) + f’‘(a)(1/2!)(x-2)^2 + f’’‘(a)(1/3!)(x-a)^3 + …
• order = p(0), p(1), p(2)…

Question 19

Binomial series (first 4 terms)

Answer 19

m = exponent of quantity
if x has coefficient x = “_x”

1 + mx + m(m-1)(1/2!)x^2 + m(m-1)(m-2)(1/3!)x^3 + …
Built off of (1 + x)^m

Question 20

Odd Numbers, Even Numbers

Answer 20

Odd: 2n + 1, Even: 2n