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Thread: Find all values of a for which the following improper converges.

  1. #1
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    Find all values of a for which the following improper converges.

    $ \Large I = \int_{0}^{\infty} e^{ax} cos(x) dx $
    I calculated the indefinite integral and founded it equals $ \large \dfrac{e^{ax}}{a^{2}+x} (sin(x)+acos(x) ) + C $
    Now, replacing $ \infty $ by t and evaluate the limit I faced this limit :
    $ \large \dfrac{1}{a^{2}+1} \lim_{t \to \infty} e^{at} \cdot (sin(t)+a cos(t)) - a ) $
    I stopped at this limit.
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  2. #2
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    Re: Find all values of a for which the following improper converges.

    That will definitely diverge unless a is negative.
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  3. #3
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    Re: Find all values of a for which the following improper converges.

    Why is that?
    ok if a is positive then e^at will be infinity, but inside brackets will be not defined as sin and cos are not exist at infinity.
    If a is negative it will be like
    $ \large \dfrac{sin(t)+acos(t)-a}{e^{kt}} $ with k=-a>0
    How to calculate this limit algebraically?
    Last edited by TWiX; Jan 28th 2018 at 09:51 PM.
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    Re: Find all values of a for which the following improper converges.

    Quote Originally Posted by TWiX View Post
    Why is that?
    ok if a is positive then e^at will be infinity, but inside brackets will be not defined as sin and cos are not exist at infinity.
    If a is negative it will be like
    $ \large \dfrac{sin(t)+acos(t)-a}{e^{kt}} $ with k=-a>0
    How to calculate this limit algebraically?
    $\begin {align*}
    &\displaystyle \int_0^\infty ~e^{a x}\cos(x)~dx = \\ \\

    &\left(\lim \limits_{x \to \infty} \dfrac{e^{a x}(a \cos(x)+\sin(x))}{1+a^2}\right) - \dfrac{e^{0}(a \cos(0)+\sin(0))}{1+a^2} = \\ \\

    &\left(\lim \limits_{x \to \infty} \dfrac{e^{a x}(a \cos(x)+\sin(x))}{1+a^2}\right) -\dfrac{a}{1+a^2}

    \end{align*}$

    if $a>0$ it should be pretty clear that the limit on the left hand side blows up.

    if $a=0$ this is the limit of a periodic function which thus doesn't exist

    if $a<0$ then this is a damped exponential which converges to 0 and thus

    $a < 0 \Rightarrow \displaystyle \int_0^\infty ~e^{a x}\cos(x)~dx = -\dfrac{a}{1+a^2}$
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  5. #5
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    Re: Find all values of a for which the following improper converges.

    What do you mean mathematically when a>0 the limit will "blow up"?
    Give me mathematical explaination here.
    for x-->infinity e^ax is infinity but we have cos(x) and sin(x) there.
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  6. #6
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    Re: Find all values of a for which the following improper converges.

    Quote Originally Posted by TWiX View Post
    What do you mean mathematically when a>0 the limit will "blow up"?
    Give me mathematical explaination here.
    for x-->infinity e^ax is infinity but we have cos(x) and sin(x) there.
    $\lim \limits_{x\to 0}~e^{a x} = \begin{cases} \infty &a > 0 \\ 1 &a=0 \\ 0 &a<0 \end{cases}$

    while it's true that the trig terms cause the value of the expression to oscillate it's still true that

    $a>0 \Rightarrow \forall \epsilon > 0,~\exists x \ni e^{ax}(a \cos(x) + \sin(x)) > \epsilon$

    i.e. the expression "blows up"
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