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Math Help - Maclaurin Series of xln(1+2x)?

  1. #1
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    Maclaurin Series of xln(1+2x)?

    Hi,

    Can someone suggest a short way of solving this? I'm doing it the regular way, that is, I took it's derivatives up to seventh degree, then put 0 into all of them, wrote a few terms from the Taylor series formula, then came up with a rather ugly general term (with 4 individiual terms before the sigma).

    Is there a better-easier way of solving this?

    Thanks.
    Last edited by Fox555; March 25th 2010 at 01:06 PM.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Is...

    \ln (1+ \xi) = \xi - \frac{\xi^{2}}{2} + \frac{\xi^{3}}{3} - \dots (1)

    ... so that all what you have to do is setting in (1) 2x instead of \xi and then multiply by x...

    Kind regards

    \chi \sigma
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  3. #3
    Super Member General's Avatar
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    Quote Originally Posted by Fox555 View Post
    Hi,

    Can someone suggest a short way of solving this? I'm doing it the regular way, that is, I took it's derivatives up to seventh degree, then put 0 into all of them, wrote a few terms from the Taylor series formula, then came up with a rather ugly general term (with 4 individiual terms before the sigma).

    Is there a better-easier way of solving this?

    Thanks.
    All what you need is knowing the Maclaurin Series for ln(1+t) ..
    Last edited by General; March 25th 2010 at 11:53 AM. Reason: typo .. *sigh*
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  4. #4
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    Ok, thanks to both of you, but after putting 2x in place of t in the expansion of ln(1+t), how am I going to multiply this by x? Shall I just write the whole sigma formula multiplied by x? Would it be OK like that?

    Sorry if this is obvious, but my mind seems to have stopped working after many hours of studying. Hope you understand.

    Thanks.
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  5. #5
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    Quote Originally Posted by Fox555 View Post
    how am I going to multiply this by x? Shall I just write the whole sigma formula multiplied by x? Would it be OK like that?
    yes, it'll be OK if you multiply whole sigma formula.
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  6. #6
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    Quote Originally Posted by Lafexlos View Post
    yes, it'll be OK if you multiply whole sigma formula.
    Ok, got it. Here is the very result then (in sigma notation) .

    xln(1+2x) =

    Right? It seems somewhat off to me though...
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  7. #7
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    if you send x to the inside of sigma, you will get something like this :

    \sum_{n=1}^\infty {(-1)}^{n-1} \frac {{2^n}{x^{n+1}}}{n}
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  8. #8
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    Thank you again Lafexlos.
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