Calc BC Unit 10: Series

Sequence vs Series

Comma vs plus

Arithmetic vs geometric

Whether it has common difference (d) or common ratio (r ).

Factorials

n!= n(n-1)(n-2)….

Bigger function order

FEPLT, For every pirate likes treasure

Sequence convergence/divergence

Depends only on the behavior of the last term (or going to infinity), not the sum

Geometric Series Test (GST)

A series diverges if |r| > or = 1

Converges if 0 < |r| < 1

If convergent, series value is equal to a/1-r, where a is first term.

NEEDS to be in the form of a*r^n to use.

Nth term test

Take the limit as n goes to infinity. If the sequence a_n doesn’t converge to 0 then series diverges (big over small or same over same). If equal to 0, CONVERGENCE CANNOT BE PROVEN and test does not apply. Must write =/ 0

Integral Test

If the series is positive, continuous and decreasing for x is greater than or equal to 1, then take the improper integral to see if both either converge or diverge. If converges, check with GST for value. Note that the value of the integral is NOT same as series, it’s either less than or greater than.

P series

1/n^p. If p > 1 it converges. If p < 1 or =1 then divergent. State comparison. If 1/n then it’s divergent harmonic series.

Direct Comparison Test

Find the convergence/divergence of closest known series. If the series is LARGER then you can prove a smaller one converges. If it is SMALLER you can prove a larger series diverges. Must indicate with either < or >

Limit Comparison Test

Take known series and take the limit of the given series divided by this known series as n goes to infinity. If the limit is >0 and finite then both series either converge or diverge.

If the series has a (-1)^n in it…

Try Alternating series first. CANNOT just go to p-series or geometric series test because it doesn’t always apply. try using something like ratio test.

Alternating Series Test

The terms of the series alternate in sign and decrease in magnitude to 0. ONLY proves convergence and you must take the limit to infinity first to prove the nth test is fulfilled and not applicable

Error bounds

Sum of x terms, error is ±value of the next term NOT included in the series. Remember ABSOLUTE VALUE BARS. If it asks for 10^-3 answers. Do plug and chug do NOT try to do it algebraically. And remember, if it is the series S_k, then error is the series value of k+1

Intervals of Convergence

Must use ratio test. |x-c| < R. Remember that center is c NOT -c. any constants in front of x you must factor out. Remember absolute values and if the fraction is (x-#)n/n then it simplifies to (x-#) bc that number is still part of the same over same

CHECK END POINTS

If a series converges, what must be true?

lim as n goes to infinity must equal 0, fulfilling nth test.

If using u-sub in integral test

CHANGE THE BOUNDS, take the limit as b goes to infinity.

n-th term test can be shown to do what

DIVERGENCE ONLY when EQUALS 0.

The series of a constant…

IS NOT THE CONSTANT, its like adding the constant to itself an infinite number of times.

Series vs sequence

Sequence is commas, series is adding. Make sure to put the +….

3^n/2^n is…

NOT SAME OVER SAME.

If |a_n| converges…

the series must also converge. but NOT the opposite

series of 1/n*cosnπ

cosπx acts as an alternating series.

With ratio test, remember…

inconclusive if =1 and remember abs value bars

Form of a Maclaurin and Taylor Series

P_n(x) = f(c)+f’(c)(x-c)+f’’(c)(x-c)²/2!+…..
…… +f⁽ⁿ⁾(c)(x-c)ⁿ/n!

If its a Maclaurin, c = 0.

n is the degree of terms you are using to approximate, i.e. second degree Taylor is one that has a (x-c)² in it.

They may also ask for the first couple nonzero terms, which would include n=0. Remember that the Taylor series is centered at x=c and is the series from n= 0 to infinity. The more terms you have, the closer the approximation is to the actual values of the function.

Things to keep in mind when constructing a Taylor Series/Polynomial

• Proper notation, like f⁽ⁿ⁾

• linkage, like remembers the series notation or writing …..+….

• Writing the series convergence proof, like AST

• nonzero terms vs nth degree

What to do if they want you to create a new Taylor series for a function you don’t know using a known Taylor series approximation?

Manipualte the known Taylor series to get the one in question. You likely have to flush out the terms and see which ones multiple, divide, add, etc to get the degree or number of terms you want

How to find the error of a Taylor/Maclaurin Series

Use either AST error or Lagrange Error Bounds

AST error bounds for maclaurin

Must give the statement that proves the series. isa convergent AST. The errors is < the absolute value of the n+1 term

Lagrange error Bound

|f(a)-P_n(a)| < | M(a-c)ⁿ⁺¹/(n+1)!|

where a is the value you want, n is the degree of the Taylor polynomial, c is the center, and M is the next highest degree derivative (f⁽ⁿ⁺¹⁾(x))

Maclaurin Series for e^x

xⁿ/n!

interval of convergence -∞ < x < ∞

Maclaurin Series for ln(x)

(-1)ⁿ⁺¹(x-1)ⁿ/n

interval of convergence 0 < x < 2

Maclaurin Series for sin(x)

(-1)ⁿx²ⁿ⁺¹/(2n+1)!
interval of convergence -∞ < x < ∞

Maclaurin series for cos(x)

(-1)ⁿx²ⁿ/(2n)!
interval of convergence -∞ < x < ∞

Maclaurin series for 1/x

(-1)ⁿ(x-1)ⁿ
interval of convergence 0 < x < 2

Maclaurin Series for 1/(1+x)

(-1)ⁿxⁿ

interval of conergence -1 < x < 1

Maclaurin Series for 1/(1-x)

xⁿ (like a geometric series)

interval of convergence -1 < x < 1