MATH 1552 Exam #2

$$\frac00,\frac{\infty}{\infty}$$

indeterminate forms where you can directly apply L’Hopital’s Rule

$$1^{\infty},0^0,\infty^0$$

indeterminate forms where the exponent makes it indeterminate, so use natural logs

$$\left(0\cdot\infty\right),\left(\infty-\infty\right)$$

indeterminate forms where you should write the common denominator as a single quotient

L’Hopital’s Rule

$$\lim_{x\to c}\frac{f\left(x\right)}{g\left(x\right)}=\lim_{x\to c}\frac{f^{\prime}\left(x\right)}{g^{\prime}\left(x\right)}=L$$

If x>0, then$\lim_{n\to\infty}x^{\frac{1}{n}}=$ ?

1

If \left|x\right|<1, then$\lim_{n\to\infty}x^{n}=$ ?

0

If \alpha>0, then$\lim_{n\to\infty}\frac{1}{n^{\alpha}}=$ ?

0

$\lim_{n\to\infty}\frac{x^{n}}{n!}=$ ?

0

$\lim_{n\to\infty}\frac{\ln\left(n\right)}{n}=$ ?

0

$\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}=$ ?

$$e^{x}$$

$\lim_{n\to\infty}n^{\frac{1}{n}}=$ ?

1

a definite integral is improper if:

• the function has a vertical asymptote at x=a, x=b, or some point c in the interval (a,b).

• one or both of the limits of integration are infinite (+∞/-∞).

if an improper integral evaluates to a finite number, we say it:

converges

if the integral evaluates to ±∞ or to ∞-∞, we say the integral:

diverges

what is a sequence?

• a sequence is a function from the set of positive integers to the set of real numbers.

• $$\left\lbrace a_{n}\right\rbrace=\left\lbrace f\left(n\right)\right\rbrace=a_1,a_2,a_3,\ldots,a_{k},\ldots$$

• $a_{n}$ is called the $n^{\operatorname{th}}$ term

• the values of n are all positive integers, unless otherwise specified.,

least upper bound (L.U.B.)

the smallest possible upper bound of a set where a number M is greater than or equal to each element in S.

greatest lower bound (G.L.B.)

the largest possible lower bound of a set where a number m is less than or equal to each element in S.

a_{n}<a_{n+1} for all n

strictly increasing sequence

$a_{n}\le a_{n+1}$ for all n

monotonically increasing sequence

a_{n}>a_{n+1} for all n

strictly decreasing sequence

$a_{n}\ge a_{n+1}$ for n

monotonically decreasing sequence

let $\left\lbrace a_{n}\right\rbrace$ be a sequence. if $\lim_{n\to\infty}a_{n}=L$, then L is the:

limit of this sequence.

if a sequence has a finite limit L, then the sequence is said to ______ to L.

converge

if a sequence does not converge to a finite limit L, it is said to:

diverge

if a sequence $\left\lbrace a_{n}\right\rbrace$ is monotonic and bounded, then it:

converges

if a sequence $\left\lbrace a_{n}\right\rbrace$ is increasing then:

L = L.U.B.

if a sequence $\left\lbrace a_{n}\right\rbrace$ is decreasing, then:

L = G.L.B.

if a sequence $\left\lbrace a_{n}\right\rbrace$ is unbounded, then it:

diverges

infinite series

a sum of infinite terms:

$$\sum_{k=0}^{\infty}a_{k}=a_0+a_1+a_2+a_3+\ldots+a_{n}+\cdots$$

an infinite series converges if:

the sequence of partial sums converges (and diverges otherwise).

$$\lim_{n\to\infty}S_{n}=\lim_{n\to\infty}\sum_{k=0}^{n}a_{k}=L$$

$$\sum_{n=1}^{\infty}\frac{1}{n}$$

harmonic series (diverges)

$$\sum_{n=1}^{\infty}\frac{1}{\left(n+a\right)\left(n+b\right)}$$

telescoping series

• these generally converge

• find the sum using partial fractions

$$\sum_{n=0}^{\infty}r^{n}$$

geometric series

when does the geometric series converge?

when |r| < 1

when does the geometric series diverge?

when |r| ≥ 1

what is the sum of a geometric sequence when |r| < 1?

$$\frac{1}{1-r}$$

$$\sum_{k=1}^{\infty}\frac{1}{k^{p}}$$

p-series

when does the p-series converge?

when p > 1

when does p-series diverge?

when 0 < p ≤ 1

divergence (n^th term test)

given $\sum_{n=0}^{\infty}a_{n}$ , first find $\lim_{n\to\infty}a_{n}$.

• if $\lim_{n\to\infty}a_{n}\ne0$ , then the series diverges

• otherwise, the test is inconclusive and you must try another test

integral test

let f be a continuous, positive, and decreasing function. then:

• $\sum_{k=1}^{\infty}f\left(k\right)$ converges if and only if $\int_1^{\infty}\!f\left(x\right)\,dx$ converges,

• and diverges if and only if $\int_1^{N}\!f\left(x\right)\,dx\to\infty$ as $N\to\infty$

justifications for the integral test

1. conditions: state that the function is positive, continuous, and decreasing

2. math step: evaluate the integral for the math step

3. conclusion: write a conclusion as to whether or not the series converges based on the result from step 2.

basic/direct comparison test, part (a)

let $\sum_{k}^{}a_{k}$ be a series with $a_{k}\ge0$ for all k. if we can find a series$\sum_{k}^{}c_{k}$ such that it converges and $a_{k}\le c_{k}$ for all but finitely many terms, then $\sum_{k}^{}a_{k}$ must also converge.

basic/direct comparison test, part (b)

let $\sum_{k}^{}a_{k}$ be a series with $a_{k}\ge0$ for all k. if we can find a series$\sum_{k}^{}d_{k}$ such that it diverges and $a_{k}\ge d_{k}\ge0$ for all but finitely many terms, then $\sum_{k}^{}a_{k}$ must also diverge.

in simpler terms, the BCT requires you to find which types of comparison series?

(a) a smaller, convergent series, OR

(b) a larger, divergent series

justifications for the BCT

1. conditions: find and state the comparison series, whether it converges/diverges, and why.

2. math step: prove, don’t just state, the inequality.

3. conclusion: summarize the results and whether the series converges or diverges.

limit comparison test

let $\sum_{k}^{}a_{k}$ be a series with $a_{k}\ge0$ for all k. select a series$\sum_{k}^{}b_{k}$. if \lim_{n\to\infty}\frac{a_{n}}{b_{n}}=c>0, then both series converge or diverge.

justifications for the LCT

1. conditions: state the comparison series, whether it converges or diverges, and why.

2. math step: find the limit of the ratio of terms. write out all the steps, don’t just “state” the limit.

3. conclusion: if the limit in step 2 is positive and finite, state that and provide a conclusion.

ratio test

let $\sum_{k=1}^{\infty}a_{k}$ be a series with all positive terms. let $L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_{n}}\right|$ .

1. if L < 1, then $\sum_{k=1}^{\infty}a_{k}$ converges

2. if L > 1, then $\sum_{k=1}^{\infty}a_{k}$ diverges

3. if L = 1, then the test is inconclusive

root test

let $\sum_{k=1}^{\infty}a_{k}$ be a series with all positive terms. let $R=\lim_{n\to\infty}\sqrt[n]{\left|a_{n}\right|}$ .

1. if R < 1, then $\sum_{k=1}^{\infty}a_{k}$ converges

2. if R > 1, then $\sum_{k=1}^{\infty}a_{k}$ diverges

3. if R = 1, then the test is inconclusive

when does a series converge by the ratio/root tests?

when L/R < 1

when does a series diverge by the ratio/root tests?

when L/R > 1

when is a series considered inconclusive by the ratio/root tests?

when L/R = 1

justifications for ratio/root tests

1. conditions: confirm and state that all terms are positive

2. math step: compute the appropriate limit, showing all mathematical steps

3. conclusion: compare your limit in step 2 to the number 1, and use that to provide a conclusion

alternating series

a series is alternating if the terms oscillate between negative and positive or vice-versa. look for: $\left(-1\right)^{k},\left(-1\right)^{k+1},\cos\left(k\pi\right)$

which test should you always start with?

the divergence/n^th term test

when should you use the integral test?

when the function looks “easy” to integrate or can be solved with a U-substitution

which kinds of series should you use in the BCT/LCT?

harmonic series, geometric series, p-series

when should you use the limit comparison test?

when you are unsure of which way the inequality may go in the BCT, or when it points in the wrong direction

when should you use the root test?

when everything is raised to the k^th power

when should you use the ratio test?

when you have factorials. or when no other test works

alternating series test

let $\sum_{n}^{}a_{n}$ be an alternating series.

1. if $\sum_{n}^{}\left|a_{n}\right|$ converges, then the alternating series converges absolutely.

2. if step 1 fails, but if:

   • {$\left|a_{n}\right|$} is a decreasing sequence, and

   • $\lim_{n\to\infty}\left|a_{n}\right|=0$, then the series converges conditionally.

3. if the terms do not go to zero, the series diverges.

4. if the terms go to zero but are not decreasing, the test is inconclusive.

when does a series converge absolutely by the AST?

when $\sum_{n}^{}\left|a_{n}\right|$ converges

when does a series converge conditionally by the AST?

if $\sum_{n}^{}\left|a_{n}\right|$ diverges, but if:

• {$\left|a_{n}\right|$} is a decreasing sequence, and

• $$\lim_{n\to\infty}\left|a_{n}\right|=0$$

when does a series diverge by the AST?

when the terms do not go to zero

what is a power series?

an infinite polynomial and a function of x

$$\sum_{k=0}^{\infty}a_{k}x^{k}$$

power series in x

$$\sum_{k=0}^{\infty}a_{k}\left(x-c\right)^{k}$$

power series in x-c

when does a power series converge?

$\sum_{k=0}^{\infty}a_{k}\left(x-c\right)^{k}$ converges at x₀ if $\sum_{k=0}^{\infty}a_{k}\left(x_0-c\right)^{k}$ converges. the series converges on (x₀, x₁) if it converges at every point in the interval.

interval of convergence

the set of all values of x of a power series for which the series converges. this interval may be closed, open, or half-open.

which tests do you use to find the radius of convergence of a power series in standard form?

ratio/root tests

when finding the R.C. of a power series, if L = 0, then:

R = ∞ and I.C. is just all real numbers

when finding the R.C. of a power series, if L = ∞, then:

R = 0 and I.C. is just x = c

when finding the R.C. of a power series, if L is positive and finite, then:

R = 1/L, and the series converges for |x - c| < R (you must also check the endpoints for convergence)

$$\frac{d}{dx}\left(\sum_{k=0}^{\infty}a_{k}x^{k}\right)=?$$

$$\sum_{k=1}^{\infty}ka_{k}x^{k-1}$$

$$\int_{}\left(\sum_{k=0}^{\infty}a_{k}x\!^{k}\,\right)dx=?$$

$$\sum_{k=0}^{\infty}\frac{a_{k}}{k+1}x^{k+1}+C$$

a taylor polynomial for a continuous function f about x = a is defined as:

$$P_{n}\left(x\right)=\sum_{k=0}^{n}\frac{f^{\left(k\right)}\left(a\right)}{k!}\left(x-a\right)^{k}$$

form of a taylor polynomial when a = 0:

$$P_{n}\left(x\right)=\sum_{k=0}^{n}\frac{f^{\left(k\right)}\left(0\right)}{k!}x^{k}$$

the taylor remainder term for P_n, where c is some number between a and x, is given by:

$$R_{n}\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}\left(x-a\right)^{n+1}$$

we can find an upper bound for the taylor remainder term using the formula:

$$\left|R_{n}\left(x\right)\right|\le\max\left|f^{\left(n+1\right)}\left(c\right)\right|\frac{\left|x-a\right|^{n+1}}{\left(n+1\right)!}$$

taylor series

$$f\left(x\right)=\sum_{k=0}^{\infty}\frac{f^{\left(k\right)}\left(a\right)}{k!}\left(x-a\right)^{k}$$

what do we call a taylor series where a = 0?

a maclaurin series

$e^{x}=?$ (maclaurin series)

$\sum_{k=0}^{\infty}\frac{x^{k}}{k!},x\in$ ℝ

$\sin\left(x\right)=?$ (maclaurin series)

$\sum_{k=0}^{\infty}\left(-1\right)^{k}\frac{x^{2k+1}}{\left(2k+1\right)!},x\in$ ℝ

$\cos\left(x\right)=?$ (maclaurin series)

$\sum_{k=0}^{\infty}\left(-1\right)^{k}\frac{x^{2k}}{\left(2k\right)!},x\in$ ℝ

$\frac{1}{1-x}$ (maclaurin series)

\sum_{k=0}^{\infty}x^{k},\left|x\right|<1

$\ln\left(1+x\right)=?$ (maclaurin series)

\sum_{k=0}^{\infty}\left(-1\right)^{k}\cdot\frac{x^{k+1}}{k+1},-1<x\le1