Calculus Unit 10

Series

What do you call an expression of the form a1+a2+a3+⋯+an+…?

Terms of a Series

In the series a1+a2+a3+… what are the individual elements a1, a2, a3,… called?

Partial Sum

For a series ∑n=1∞an, what is the sum Sn=a1+a2+⋯+an known as?

Sum of a Series

If the sequence of partial sums Sn converges to a limit S, then S is called the ______ of the series.

Convergent Series

A series is called ______ if its sequence of partial sums converges.

Divergent Series

A series is called ______ if its sequence of partial sums does not converge.

Geometric Series

What do you call a series of the form a+ar+ar2+⋯+arn−1+…?

Common Ratio

In a geometric series a+ar+ar2+…, what is the value r called?

Convergence of a Geometric Series

A geometric series with ratio r converges if ______ and its sum is ______.

Power Series

What do you call a series of the form ∑n=0∞cn(x−a)n=c0+c1(x−a)+c2(x−a)2+…?

Radius of Convergence

For a power series, what is the number R≥0 such that the series converges for |x−a|R called?

Interval of Convergence

What do you call the set of x-values for which a power series converges?

Taylor Series

If a function f has derivatives of all orders at x=a, then the series ∑n=0∞f(n)(a)n!(x−a)n is called the ______ series of f at a.

Maclaurin Series

What is the Taylor series of a function f centered at a=0 called?

Taylor Polynomial of Order n at a

What do you call the polynomial Pn(x)=f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+⋯+f(n)(a)n!(x−a)n that approximates f(x) near x=a?

Remainder of Order n (Taylor’s Formula)

If f(x)=Pn(x)+Rn(x), then Rn(x) is called the ______.

Taylor’s Theorem with Remainder

State the formula that expresses the remainder Rn(x) in terms of an integral or a derivative evaluated at an unknown point.

Telescoping Series

What type of series is one where most terms cancel out in the partial sums, leaving only a fixed number of terms?

Alternating Series Test

What test can be used to determine the convergence of a series whose terms alternate in sign?

Absolute Convergence

A series ∑an converges ______ if the series of absolute values ∑|an| converges.