Sequences and Series & Taylor series Patrick Doyle Elwin Martin Charles Ye

Series and Sequences Sequence: function whose domain is a set of all positive integers (n=1,2,3…) 2

Limit of a Series If the sequence converges, then

example Series diverges Series converges

nth term test If   o or if the limit does not exist, then the series diverges. Apples to any series Cannot be used to prove convergence

example Not enough information Series diverges

Geometric series Converges if the absolute value of r is less than one Diverges if the absolute value of r is greater than or equal to one Where a is the first time

example

Alternating harmonic series

Harmonic series Always diverges

Telescoping series Series where the later terms start to cancel out Converges to the numbers that do not cancel out

example Converges to 1

Integral Test Converges only if converges Diverges only if diverges Where the function is continuous, positive, and decreasing

example

P-series test If p is greater than one, the series converges If p is equal or less than one, the series diverges

Direct Comparison Test Converges if and converges Diverges if and diverges

example

Limit Comparison Test If is a number greater than zero and converges, then converges If is a number greater than zero and diverges, then diverges -Both series a and b must be greater than zero - The limit must not be infinity, zero, or negative

Root Test There exists r such that if r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

example

Ratio Test There exists a number r such that: if r < 1, then the series converges; If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

Error in Alternating series

Radius and Interval of Convergence In some instances there are series written in terms of x such as and in such cases there are a range of x values for which the series converges. The radius of curvature and interval of convergence are found by applying the ratio test and setting the ratio to less than one and solving for x. Once the interval has been found, check the end points.

example

Power Series Representation of a function as a summation of infinitely many polynomials. C= x value at which the series is centered

Finding the Coefficient I FOUND IT!!! Using this, you can plug it in for several terms then find a the equation for the general term! Now you have a Taylor Series!

(Actually finding the coefficient)

Example Find the first 3 terms and the general term of the Taylor series for sin(x) centered at 0. f (x) = sin(x) f (0) = 0 f '(x) = cos(x) f '(0) = 1 f '' = -sin(x) f ''(0) = 0 f ''' = -cos (x) f '''(0) = -1 f '''' = sin(x) f ''''(0) = 0 f''''‘=cos(x) f''''‘(0)=1

Manipulating the Series You can tweak the formulas you memorized to make finding the Maclaurin Series for many more equations much faster and EASIER. So do that. You can multiply the whole series by a number, substitute for x, or take the integral or derivative.

Maclaurin Series A Maclaurin series is a Taylor Series that centers at 0 (i.e. c=0). Here are some to memorize:

Example Find the first four terms of the Maclaurin series for 5e2x.

Lagrange Error Bound Where f(x) is centered at c and a is some number between x and c.