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  1. #1
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    pls help

    $\displaystyle S_1 = \sum_{k = 1}^n k cos k$

    $\displaystyle S_2 = \sum_{k = 1}^n k sin k$

    $\displaystyle S_1, S_2 = ? using: S_1 + iS_2$
    pls help
    thanx
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  2. #2
    Newbie hurrem's Avatar
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    Quote Originally Posted by ely_en View Post
    $\displaystyle S_1 = \sum_{k = 1}^n k cos k$

    $\displaystyle S_2 = \sum_{k = 1}^n k sin k$

    $\displaystyle S_1, S_2 = ? using: S_1 + iS_2$
    pls help
    thanx
    What I understand from S1 is that is is $\displaystyle S_1 = \sum_{k = 1}^n cos k$ = cos1+cos2+cos3+cos4+ .... +cos(n-1)+cosn which is kx{[cos(n)]x[cos(n+1)]/2}

    and the same thing for S2 (Im assuming) and then adding up as in x+iy form.

    Hope that helped...
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  3. #3
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    $\displaystyle S_1 = \sum_{k = 1}^n k*cos k, not \sum_{k = 1}^n cos k$
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  4. #4
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    Hello, ely_en!

    I haven't solved it yet . . . but I ran across something interesting.


    $\displaystyle S_1 \:= \:\sum_{k=1}^n k\cos k$

    $\displaystyle S_2 \:=\: \sum_{k=1}^n k\sin k$

    Find $\displaystyle S_1,\:S_2$ . . . using: $\displaystyle S_1 + iS_2$

    . . . $\displaystyle S_1 \;=\;1\cos1 + 2\cos2 + 3\cos3 + 4\cos4 + \hdots$
    . . . $\displaystyle iS_2 \:=\:i\sin1 + 2i\sin2 + 3i\sin3 + 4i\sin4 + \hdots$

    Add: .$\displaystyle S_1+iS_2 \;=\;(\cos1+i\sin1) +2(\cos2 + i\sin2) + 3(\cos3 + i\sin3)$ $\displaystyle + 4(\cos4+i\sin4) + \hdots$


    And we have: . . . $\displaystyle S_1+iS_2 \;=\;e^i + 2e^{2i} + 3e^{3i} + 4e^{4i} + \hdots\;\;{\color{blue}[1]}$

    Multiply by $\displaystyle e^i\!:\;\;e^i(S_1 + iS_2) \;=\;\qquad \;e^{2i} + 2e^{3i} + 3e^{4i} + \hdots\;\;{\color{blue}[2]}$


    $\displaystyle \text{Subtract }{\color{blue}[1] \text{ - } [2]}\!:\;\;(1-e^i)(S_1+iS_2) \;=\;\underbrace{e^i+ e^{2i} + e^{3i} + e^{4i} + \hdots}_{\text{geometric series}}$

    $\displaystyle \text{Hence: }\;(1-e^i)(S_1+iS_2) \;=\;\frac{e^i}{1-e^i}$

    $\displaystyle \text{Therefore: }\;S_1+iS_2 \;=\;\frac{e^i}{(1-e^i)^2} $


    A fascinating result . . .
    . . absolutely useless, but still fascinating.

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  5. #5
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    Wink

    thanx
    I've tried this too, but I used the moivre's formula and then the formula for the sum of a geometric progression.
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  6. #6
    MHF Contributor Mathstud28's Avatar
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    Lemma: $\displaystyle \sum_{n=1}^{N}nx^n=\frac{x^{N+1}\left(xN-N-1\right)+x}{(x-1)^2}$

    $\displaystyle \therefore\sum_{n=1}^N{n\cos(n)}$


    $\displaystyle =\text{Re}\bigg[\sum_{n=1}^N{n\left(e^i\right)^n}\bigg]$


    $\displaystyle =\text{Re}\bigg[\frac{\left(e^i\right)^{N+1}\left(e^iN-N-1\right)+e^i}{(e^i-1)^2}\bigg]$

    $\displaystyle =\frac{\left((\cos(1)-1)N-1\right)\cos\left(N\right)-\sin(1)N\sin\left(N\right)+1}{2\left(\cos(1)-1\right)}$

    Do the same except its

    $\displaystyle \sum{n\sin(n)}=\text{Im}\bigg[\sum{n\left(e^i\right)^n}\bigg]$
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  7. #7
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Soroban View Post

    $\displaystyle \text{Subtract }{\color{blue}[1] \text{ - } [2]}\!:\;\;(1-e^i)(S_1+iS_2) \;=\;\underbrace{e^i+ e^{2i} + e^{3i} + e^{4i} + \hdots}_{\text{geometric series}}$

    $\displaystyle \text{Hence: }\;(1-e^i)(S_1+iS_2) \;=\;\frac{e^i}{1-e^i}$
    Hmm, you seemed to apply the rule $\displaystyle \sum_{n=1}^{{\color{red}{\infty}}}x^n=\frac{x}{1-x}$


    But this series is clearly divergent as $\displaystyle N\to\infty$

    Besides...the OP did not ask for the infinite series.

    Hope I have made sense,

    Mathstud.
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  8. #8
    MHF Contributor Mathstud28's Avatar
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    Since it doesnt appear the OP isn't going to post the answer to $\displaystyle \sum_{n=1}^{N}n\sin(n)$ I will present it

    $\displaystyle \begin{aligned}
    \sum_{n=1}^{N}n\sin(n)&=\text{Im}\bigg[\sum_{n=1}^{N}n\left(e^i\right)^n\bigg]\\
    &=\text{Im}\bigg[\frac{\left(e^i\right)^{N+2}N-\left(e^i\right)^{N+1}N-\left(e^i\right)^{N+1}+e^i}{\left(e^i-1\right)^2}\bigg]\\
    &=\frac{\sin(1)N\cos\left(N\right)+\cos(1)N\sin\le ft(N\right)-N\sin\left(N\right)-\sin\left(N\right)}{2\left(\cos(1)-1\right)}
    \end{aligned}$
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