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Thread: Calculus Multiple Choice

  1. #1
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    Exclamation Calculus Multiple Choice

    Mutliple Choice questions

    5. Determine which converges:

    a. $\displaystyle \sum^{\infty}_{n=1}(4+(-1)^n)$.
    b. $\displaystyle \sum^{\infty}_{n=1}2^n/(n+1)!$.
    c. $\displaystyle \sum^{\infty}_{n=0} 5(3/2)^n$.
    d. $\displaystyle \sum^{\infty}_{n=1} 1/n^(1/2)$.
    e. None of these

    7. Determine which test can be proved the divergence $\displaystyle \sum^{\infty}_{n=1} 1/n^(1/2)$.

    a. Geometric Series Test
    b. P Series Test
    c. Ratio Test
    d. Nth Term Test for Divergence
    e. None of These

    14. Determine which of the following test could be used to show that $\displaystyle \sum^{\infty}_{n=1} (2n-1)/(3n+5)$.


    a. Root Test
    b. Ratio Test
    c. Geometric Series Test
    d. P-Series Test
    e. None of These

    17. Find the radius of convergence of the power series $\displaystyle \sum^{\infty}_{n=0} (x/2)^n$.

    a. 1/2
    b. 2
    c. Infinity
    d. 0
    e. None of These
    Last edited by Nimmy; Feb 23rd 2006 at 03:47 PM.
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  2. #2
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    Problem 5:
    a)This diverges by oscillation.
    b)This converges.
    c)This is a geometric series with the constant ration $\displaystyle |3/2|>1$ thus it diverges.
    d)This is called the p-series, since the exponent is less than 1 it must diverge.

    Explaination to b:
    Apply the ratio test,
    $\displaystyle a_{k+1}=\frac{2^{k+1}}{(k+2)!}$
    $\displaystyle a_k=\frac{2^k}{(k+1)!}$
    Thus,
    $\displaystyle \lim_{k\to\infty}\frac{2^{k+1}}{(k+2)!}\frac{(k+1) !}{2^k}=\frac{2}{k+2}=0<1$
    Thus, it converges.
    Q.E.D.
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  3. #3
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    Exclamation

    Quote Originally Posted by Nimmy
    Mutliple Choice questions

    5. Determine which converges:

    a. $\displaystyle \sum^{\infty}_{n=1}(4+(-1)^n)$.
    b. $\displaystyle \sum^{\infty}_{n=1}2^n/(n+1)!$.
    c. $\displaystyle \sum^{\infty}_{n=0} 5(3/2)^n$.
    d. $\displaystyle \sum^{\infty}_{n=1} 1/n^(1/2)$.
    e. None of these

    7. Determine which test can be proved the divergence $\displaystyle \sum^{\infty}_{n=1} 1/n^(1/2)$.

    a. Geometric Series Test
    b. P Series Test
    c. Ratio Test
    d. Nth Term Test for Divergence
    e. None of These

    14. Determine which of the following test could be used to show that $\displaystyle \sum^{\infty}_{n=1} (2n-1)/(3n+5)$.


    a. Root Test
    b. Ratio Test
    c. Geometric Series Test
    d. P-Series Test
    e. None of These

    17. Find the radius of convergence of the power series $\displaystyle \sum^{\infty}_{n=0} (x/2)^n$.

    a. 1/2
    b. 2
    c. Infinity
    d. 0
    e. None of These

    I still need help on 7,14,17.
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  4. #4
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    Quote Originally Posted by Nimmy
    I still need help on 7,14,17.
    7)That is the p-series, because it has form $\displaystyle \frac{1}{k^n}$.

    14)None of these, you may solve it using a different test. The limit comparison test.

    17)Use the ratio test, $\displaystyle a_k=(x/2)^k$ and $\displaystyle a_{k+1}=(x/2)^{k+1}$.
    Thus,
    $\displaystyle \lim_{k\to\infty}\frac{x^{k+1}}{2^{k+1}}\frac{2^k} {x^k}=\frac{x}{2}$ Now it is necessary for this is converge when $\displaystyle |\frac{x}{2}|<1$ that happens when $\displaystyle -2<x<2$ thus the radius of convergence is 2.
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  5. #5
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    Hi Folks:

    Are we doing homework or completing take-home tests for students of dimished ambition? ...just a thought (not an insinuation).

    Respectfully,

    Rich B.
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