BC Test
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School
University of California, Los Angeles *
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Course
241
Subject
Mathematics
Date
Feb 20, 2024
Type
md
Pages
4
Uploaded by MasterBraveryPorcupine32
### True or False Questions
1. The Harmonic Series $\sum_{n=1}^{\infty} \frac{1}{n}$ converges
(T / F)
2. The Alternating Harmonic Series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges (T / F)
### Multiple Choice Questions
1. Which of the following Series converge?
$I. \sum_{n=1}^{\infty}\frac{8^n}{n!}$ $II. \sum_{n=1}^{\infty}\frac{n!}
{n^{100}}$ $III. \sum_{n=1}^{\infty}e^{-n}(n^2+4)$
a) I only
b) II only
c) I and III only
d) I, II, and III 2. What does the series $\sum_{n=1}^{\infty}\frac{4n+2}{3n}$ converge to?
a) $\frac{4}{3}$
b) 1
c) 0
d) Divergent
3. What is the value of $S_4$ in the sequence of partial sums $S_n$ of sequence $a_n = \frac{\pi^n}{2^n}, n>0$
a) $\frac{\pi^4}{16}$
b) $\frac{\pi^4+2\pi^3+4\pi^2+8\pi}{16}$
c) $\frac{\pi^4+2\pi^3+4\pi^2+8\pi}{32}$
d) $\frac{\pi^4+2\pi^3}{8}$
4. If the $\lim_{n\rightarrow\infty} S_n = 1$ where $S_n$ is the sequence of partial sums of the sequence $a_n$. What does $\sum_{n=1}^{\infty}a_n$ equal?
a) 1
b) 0
c) $\infty$
d) Can not be determined
5. Considering the Alternating Series $\sum_{n=0}^{\infty}(-1)^na_n$ which of the following are true?
$I.$ If the series converges conditionally, any rearranged order of the series sums to the same value
$II. $ If the series converges absolutely, any rearranged order of the series sums to the same value
$III. $ If the series $\sum_{n=0}^{\infty}a_n$ converges, then $\sum_{n=0}^{\
infty}(-1)^na_n$ converges
a) I only
b) II only
c) II and III only
d) I and III only
### Free Response Questions
1. All questions in this section refer to the series $\sum_{n=0}^{\infty} \frac{(-
1)^n}{2n+1}$
a) Determine whether the given series converges or not
b) Determine the amount of terms the series above must be summed to, in order to
acheive a remainder less than $10^{-6}$ given the exact value of the series $\
frac{\pi^3}{32}$ (Although the exact value is not needed to solve the question)
c) Find an upper bound of the magnitude of the error in approximating the value of the series with $n = 8$ terms given the exact value of the series $\frac{\pi^3}
{32}$ (Although the exact value is not needed to solve the question)
2. Given the series $\sum_{n=1}^{\infty} \frac{1}{e^{n}}$
a) Determine whether the given series converges, if it does find the value $S$ of the series.
b) State the conditions for the Integral Test, if the conditions are met use the
Integral test to show $\sum_{n=1}^{\infty} \frac{1}{e^n}$ converges
c) Use the given series and the limit comparison test to show $\sum_{n=1}^{\
infty} \frac{1}{2e^n+3}$ converges
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3. Given the series $\sum_{n=1}^{\infty} \frac{(-1)^n \ln(n)}{n^3}$ a) Prove whether the series converges or diverges without using the Alternating Series Test ### Investigative Task
The Fractal called the *Sierpinksi Triangle* is the limit of a sequence of figures.
Starting with the equilateral triangle with sides of length 1, an inverted equilateral triangle with sides of length $\frac{1}{2}$ is removed. Then, three inverted triangles with side length $\frac{1}{4}$ are removed from this figure. The
process continues in this way. Let $T_n$ be the total area of the removed triangles after stage $n$ of this process. The area of an equilateral triangle with
side lentgh $L$ is $A = \frac{\sqrt{3}L^2}{4}$
a) Find $T_1$ and $T_2$ b) Find an explicit formula for $T_n$
c) Find $\lim_{n\rightarrow\infty}T_n$
d) What is the area of the original triangle that remains as $n\rightarrow\infty$ ?