Sequence: An infinite succession of numbers that follow a pattern
Officially:
Infinite Series
An infinite series is an expression of the form a₁ + a₂ + a₃… + aₖ + …. The numbers a₁, a₂, a₃, etc. are the terms of the series
As n increases, the partial sum includes more and more terms of the series. So if aₙ approaches a limit as n approaches ∞, then this limit is likely to be the sum of all of the terms in the series, thus a convergence.
Used in the form of ∞Σ(n=1) a*r^(n-1)
For example, the series (1/2)+(1/2^2)+(1/2^3) converges, while 2+2^2+2^3 diverges.
If a geometric series converges, you can figure its sum out by doing the following:
If we want to find the sum of the first n terms of a geometric series, we use the following formula:
For an infinite geometric series, if it converges, lim (n→∞) r^n = 0, and its sum is found this way:
The nth Term Test For Divergence
The nth term test means we find the limit of the series and see if we get zero or not
Integral Test for Convergence
Either both converge or both diverge.
In other words, if you want to test convergence for a series with positive terms, evaluate the integral and see if it converges.
Harmonic Series: (1/n) pattern
p-Series
Comparison Tests for Convergence
Limit Comparison Test
Alternating Series Test for Convergence
Ratio Test for Convergence
Determining Absolute or Conditional Convergence
Alternating Series Error Bound
Finding Taylor Polynomial Approximations of Functions
We can approximate a function on an interval centered at x=a by using a Taylor series (if centered about x=0, it’s a Maclurin series).
In a Taylor polynomial, the terms are found from thee derivatives of f as follows:
Lagrange Error Bound
The “error bound” of a Taylor polynomial, aka the Lagrange error
If you are finding an nth degree Taylor polynomial, a good approximation to the error bound is the next nonzero term in a decreasing series.
Radius and Interval of Convergence of Power Series
Representing Functions as Power Series