AP Calculus BC Unit 10 infinite sequences and series

Intro

Infinite sequence - an infinite succession of numbers: 1, 2, 3, 4,; or 1/2, 1/3, 1/4, …

Infinite series - the sum of the terms of an infinite sequence: 1 + 2 + 3 + 4+…; or ½ + 1/3 + ¼ +…; $\sum$ a_n

Partial sum - the sum of the first n terms of a series: S_n = a₁ + a₂ + a₃ + … + a_n

If the series of partial sums has a limit L as n approaches infinity, the series converges to L.

If the series doesn’t have a finite limit, the series diverges and has no sum.

Convergence tests

nth term test - if the limit of a_n does not = 0 as n approaches infinity, then sum to infinity a_n will diverge. This test doesn’t confirm convergence, it can only tell you that it will diverge. If the limit doesn’t equal 0, the test is inconclusive (as the series could be oscillating, or something that causes it to not approach a single limit.

integral test - the integral of a_n from 1(or where n starts) to infinity will converge or diverge with the sum to infinity (though they will not necessarily equal the same thing.) if other tests don’t work out and the function is integrable, this test is a great option!

comparison test - when a_n <= b_n for all n, if b converges, then a converges. if a diverges, then b diverges. In other words, if a larger function converges then the small one will too. if the smaller function diverges then the larger one will too.

choose a b_n to compare to that is similar in form to a_n and we know is larger/smaller, and converges/diverges

limit comparison test - take the limit to infinity of a_n/b_n

if the limit = c (c>0) then both converge or both diverge.

if the limit = 0 and b_n converges then a_n also converges.

if the limit = infinity and b_n diverges then a_n also diverges.

if you see a series that is a ratio of polynomials, you can often use the limit comparison test.

alternating series test -

if the series meets all 3 criteria, then the series will converge.

1. the series is in the form $\sum$ (-1)ⁿ a_n or (-1)ⁿ⁺¹ a_n

2. a_n < a_n+1 (is decreasing)

3. lim n →infinity a_n = 0

ratio test -

divide the (n+1)th term by the nth term to find the common ratio. as n approaches infinity, the ratio should approach a number.

If the ratio < 1, the series converges.

If the ratio > 1, the series diverges.

If the ratio = 1, the test is inconclusive (it could diverge or converge, use another test).

absolute convergence -

If both |a_n| converges and a_n converges, it converges absolutely.

example, (-1)ⁿ1/n² converges, and the absolute value of it converges.

conditional convergence -

If a_n converges but |a_n| diverges, it converges conditionally.

example: (-1)ⁿ⁺¹1/n converges, but |(-1)ⁿ⁺¹1/n| diverges.

Noteworthy types of series

Geometric series - have a common ratio between each term (each successive term is r times the last term). If the ratio is less than 1, the series converges. If the ratio is greater than or equal to 1, the series diverges.

a geometric series $\sum$ a(rⁿ) converges to a/(1-r), if it converges.

harmonic series - a series in the form $\sum$ 1/n = ½ + 1/3 + ¼ + 1/5 +…

it does not converge, because the terms are not decreasing quickly enough to converge, so the value of the harmonic series is $\infty$

alternating series -a series where each term alternates in sign. they can be in the form $\sum$ $(-1)^n$a_n or $\sum$(-1)ⁿ⁺¹ a_n

p series -

a series in the form $\sum$ $1/n^p$ = 1+ $1/2^p$ + $1/3^p$ + $1/4^p$ + …

it converges if p > 1 and diverges if 0 < p <= 1

power series -

Taylor series -

$p(x) = f(a) + f’(a)(x-a)+f’’(a)(x-a)²/2!+f’’’(a)(x-a)³/3!+…$

• $f^(n)(a)(x-a)^n/n!$ where f^(n) is the nth derivative of f

A taylor polynomial can approximate a function using a polynomial on an interval centered around x=a. the more terms of the polynomial, the better the approximation.

Mclaurin series -

a taylor series centered around 0.

• $f^(n)(0)(x)^n/n!$

Other important ideas

Alternating Series Error Bound - the error (or the difference between the partial sum S_n and the infinite sum S) of an alternating series is difficult to calculate, but you can put a bound on the error, or say that it is less than some amount. This bound is the absolute value of the term after the last one taken (so if you take a sum to 7 terms, the error will be less than the absolute value of the 8th term). This only works with alternating series

Lagrange Error Bound -

memorize the formula. the error is the remainder.

R <= $f^(n+1)(c) * (x-a)^(n+1) / (n+1)!$

(n+1)th derivative of f, (x-a) to the (n+1)th power (formatting sucks.)