Calculus Unit 10

Flashcard 1 (Term): Series
Flashcard 1 (Answer): What do you call an expression of the form a1+a2+a3+⋯+an+…a_1 + a_2 + a_3 + \dots + a_n + \dotsa1​+a2​+a3​+⋯+an​+… ?

Flashcard 2 (Term): Terms of a Series
Flashcard 2 (Answer): In the series a1+a2+a3+…a_1 + a_2 + a_3 + \dotsa1​+a2​+a3​+…, what are the individual elements a1,a2,a3,…a_1, a_2, a_3, \dotsa1​,a2​,a3​,… called?

Flashcard 3 (Term): Partial Sum
Flashcard 3 (Answer): For a series ∑n=1∞an\sum_{n=1}^{\infty} a_n∑n=1∞​an​, what is the sum Sn=a1+a2+⋯+anS_n = a_1 + a_2 + \dots + a_nSn​=a1​+a2​+⋯+an​ known as?

Flashcard 4 (Term): Sum of a Series
Flashcard 4 (Answer): If the sequence of partial sums SnS_nSn​ converges to a limit SSS, then SSS is called the ______ of the series.

Flashcard 5 (Term): Convergent Series
Flashcard 5 (Answer): A series is called ______ if its sequence of partial sums converges.

Flashcard 6 (Term): Divergent Series
Flashcard 6 (Answer): A series is called ______ if its sequence of partial sums does not converge.

Flashcard 7 (Term): Geometric Series
Flashcard 7 (Answer): What do you call a series of the form a+ar+ar2+⋯+arn−1+…a + ar + ar^2 + \dots + ar^{n-1} + \dotsa+ar+ar2+⋯+arn−1+… ?

Flashcard 8 (Term): Common Ratio
Flashcard 8 (Answer): In a geometric series a+ar+ar2+…a + ar + ar^2 + \dotsa+ar+ar2+…, what is the value rrr called?

Flashcard 9 (Term): Convergence of a Geometric Series
Flashcard 9 (Answer): A geometric series with ratio rrr converges if ______ and its sum is ______.

Flashcard 10 (Term): Power Series
Flashcard 10 (Answer): What do you call a series of the form ∑n=0∞cn(x−a)n=c0+c1(x−a)+c2(x−a)2+…\sum_{n=0}^{\infty} c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \dots∑n=0∞​cn​(x−a)n=c0​+c1​(x−a)+c2​(x−a)2+… ?

Flashcard 11 (Term): Radius of Convergence
Flashcard 11 (Answer): For a power series, what is the number R≥0R \geq 0R≥0 such that the series converges for ∣x−a∣<R|x-a| < R∣x−a∣<R and diverges for ∣x−a∣>R|x-a| > R∣x−a∣>R called?

Flashcard 12 (Term): Interval of Convergence
Flashcard 12 (Answer): What do you call the set of xxx-values for which a power series converges?

Flashcard 13 (Term): Taylor Series
Flashcard 13 (Answer): If a function fff has derivatives of all orders at x=ax = ax=a, then the series
∑n=0∞f(n)(a)n!(x−a)n\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n∑n=0∞​n!f(n)(a)​(x−a)n
is called the ______ series of fff at aaa.

Flashcard 14 (Term): Maclaurin Series
Flashcard 14 (Answer): What is the Taylor series of a function fff centered at a=0a = 0a=0 called?

Flashcard 15 (Term): Taylor Polynomial of Order nnn at aaa
Flashcard 15 (Answer): 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)nP_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots + \frac{f^{(n)}(a)}{n!}(x-a)^nPn​(x)=f(a)+f′(a)(x−a)+2!f′′(a)​(x−a)2+⋯+n!f(n)(a)​(x−a)n
that approximates f(x)f(x)f(x) near x=ax = ax=a?

Flashcard 16 (Term): Remainder of Order nnn (Taylor’s Formula)
Flashcard 16 (Answer): If f(x)=Pn(x)+Rn(x)f(x) = P_n(x) + R_n(x)f(x)=Pn​(x)+Rn​(x), then Rn(x)R_n(x)Rn​(x) is called the ______.

Flashcard 17 (Term): Taylor’s Theorem with Remainder
Flashcard 17 (Answer): State the formula that expresses the remainder Rn(x)R_n(x)Rn​(x) in terms of an integral or a derivative evaluated at an unknown point.

Flashcard 18 (Term): Telescoping Series
Flashcard 18 (Answer): What type of series is one where most terms cancel out in the partial sums, leaving only a fixed number of terms?

Flashcard 19 (Term): Alternating Series Test
Flashcard 19 (Answer): What test can be used to determine the convergence of a series whose terms alternate in sign? (Hint: State the two conditions that must be met.)

Flashcard 20 (Term): Absolute Convergence
Flashcard 20 (Answer): A series ∑an\sum a_n∑an​ converges ______ if the series of absolute values ∑∣an∣\sum |a_n|∑∣an​∣ converges.