series

series vs a sequence

→sequence: numbers with a rules which allows you to get the next number

→ series: is the sum of the numbers in a sequence

what should you always do when tackling these questions

always write out the first few terms of the series

converging vs diverging

→converging: series approaches a finite number as it goes to infinity

→ diverging: series never approaches a specific number

what is an arithmetic series:

→ how do you find the next term/general equation

→ how do you calculate the sum

→ what is it’s properties

→ series where the difference between consecutive numbers is constant

$$a_n = a_1+(n-1)d$$

$sum = \frac{n}{2}[a_1+a_N]$ remember you can substitute the equation for $a_N$ into the equation

→ the series always diverges

what is a geometric series

→ sum?

→ general expression/calculate n term

→ series where the ratio between consecutive terms is constant

$$a_N = a_{1 or 0}( r )^{n-1}$$

→ sum of an infinite series can be calculated if the ratio < 1 if it is larger than 1 you cannot calculate it

$$infinite sum = \frac{a_1}{1-r}$$

→ sum of a finite series

$$finite sum = \frac{a_1(1-r^n)}{1-r}$$

wha

what is an alternating series

$$(-1)^{n+1} a_n$$

when does an alternating serie converge

cnverges if:

a_{N+1} < a_N and $\lim_{n\to\infty}{}a_{n}=0$

what is a harmonic series and what are it’s properties?

$\frac{1}{n}$ series diverges w

what is a telescoping series

$\frac{1}{n}-\frac{1}{n+1}$ w

what are the tests for convergence and divergence?

first write out the first few terms or derive the general equation

1. calculate the limit at infinity

2. compare to a geometric series

3. compare to a P series

4. d’alembert’s ratio

5. compare to arithmetic series

   1. compare to harmonic series

What do the results of evaluating at infinity tell you about convergence or divergence

If = 0 need further testing if not the series is divergent

explain the comparing to geometric series

identify if the series has a general equation $a_1 ( r )^{n+1}$

→ if |R| > 1 series is diverging

→ if |R| < 1 series is converging

comparing to a P series

identify if the series has a general equation $\frac{1}{n^p}$

→ if P > 1 series converges

→ if P <= 1 series diverges

d’alembert’s test

find the general equation for the series

substitute n with n+1

evaluate the limit of $\frac{n+1}{n}$

→ if limit < 1 series converges

_> if limit > 1 series diverges

→ if limit = 1 test is inconclusive

direct comparison tests

→ arithmetic series always diverges therefore if series > arithmetic → diverges

→ harmonic series always diverges therefore if series > $\frac{1}{n}$ → diverges

taylor expansion

$$f(A) + f(A)’(x-a) + f(a)’’\frac{(x-a)^2}{2!}+ f(a)’’’\frac{(x-a)^3}{3!}+…+f(a)^{’n}\frac{(x-a)^n}{n!}$$

maclaurin expansion

it is a taylor expansion were a = 0 therefore

$$f(0) + f(0)’x + f(0)’’\frac{x^2}{2!}+ f(0)’’’\frac{x^3}{3!}+…+f(0)^{’n}\frac{x^n}{n!}$$

when do you need to use the power series?

when you are asked to integrate sin or cos terms which are to the power of something