Results 1 to 7 of 7

Math Help - Alternating Series

  1. #1
    Senior Member
    Joined
    Apr 2006
    Posts
    401

    Alternating Series

    "Let S = summation((-1)^i*1/[i*ln(i)-15], i = 10...infinity)

    a.) Determine whether S diverges, converges conditionally, or converges absolutely.
    b.) If S converges (whether conditionally or absolutely), find upper and lower bounds on S. If it diverges, find N so that S_N is >= 10.
    c.) Let T represent the associated series consisting of all positive terms. That is, if S = summation((-1)^i*c_i, i = 10...infinity), let
    T = summation(c_i, i = 10...infinity). If S converges absolutely, find N so that T_N is within .001 of T. If S does not converge absolutely, find N so that T_N >= 10."

    This is what I did:

    Since it's an alternating series, I can apply the alternating series test (AST). If the limit of the positive portion of it = 0, the servies S converges. That is, if limit as i -> infinity of 1/[i*ln(i) - 15] = 0, which it does. In order to determine whether it converges conditionally or absolutely, I need to take the absolute value of the series and determine if that converges or diverges.

    Ok, if the absolute value of the series converges, then the series S converges absolutely. If it does not, that is, if it diverges, then the series S converges conditionally. To determine if the absolute value of the series converges/diverges, I need to use a positive series test. I decided to use a comparison test.

    I compared 1/[i*ln(i) - 15] to 1/[i*ln(i)]:

    1/[i*ln(i)] < 1/[i*ln(i) - 15]. Now I can do an integral test on 1/[i*ln(i)]...when doing that all out (im not going to list what I did because it'd take too long), we find out that 1/[i*ln(i)] diverges. Therefore, 1/[i*ln(i) - 15] must diverge too. Thus, it converges conditionally.

    b.) I have to now find upper and lower bounds on S. Since S is an alternating series, as said before, I know S is between and two consecutive partial sums. I did S_10 and S_11. I concluded .0367 <= S <= .1246.

    c.) I am stuck on this part. I don't really know what it's asking.

    "c.) Let T represent the associated series consisting of all positive terms. That is, if S = summation((-1)^i*c_i, i = 10...infinity), let
    T = summation(c_i, i = 10...infinity). If S converges absolutely, find N so that T_N is within .001 of T. If S does not converge absolutely, find N so that T_N >= 10."

    Obviously, now that I know it converges conditionally, I have to find an N so that T_N >= 10.

    Help would be greatly appreciated, as always.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    You will need to check this carefully!

    Last part:

    <br />
T_N=\sum_{n=10}^{N} \frac{1}{n\log(n)-15}<br />

    Now:

    <br />
n\log(n)-15<n\log(n)<br />

    So:

    <br />
T_N>\sum_{n=10}^{N} \frac{1}{n\log(n)}>\int_{9}^{N-1}\frac{1}{x\log(x)}dx=\log \left( \frac{\log(N-1)}{\log(9)} \right)<br />

    So now you need only find N such that:

    <br />
 \log \left(\frac{\log(N-1)}{\log(9)}\right) >10<br />

    Which suggests that N > e^{\log(9)e^{10}}+1 will suffice

    RonL
    Last edited by CaptainBlack; May 12th 2006 at 09:21 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Apr 2006
    Posts
    401
    log? or did you mean ln?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by AfterShock
    log? or did you mean ln?
    Default meaning of log is natural log, a text will make it clear that
    it means common log if that is the default meaning in the text.

    So where I have writen log I mean natural log, so you can replace
    log by ln throughout what I have writen if that is the convention
    you are used to.

    RonL

    PS

    <br />
 e^{\log(9)e^{10}} \approx 3.89 \times 10^{21018}<br />

    Impressive!!
    Last edited by CaptainBlack; May 12th 2006 at 09:19 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Oct 2005
    From
    Earth
    Posts
    1,599
    In American schools we learn log to mean base 10 and ln to mean base e. For what reason we do this I don't know. Most higher level math literature I've read always uses log to mean base of e.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Jameson
    In American schools we learn log to mean base 10 and ln to mean base e. For what reason we do this I don't know. Most higher level math literature I've read always uses log to mean base of e.
    and almost every programming language have log being the natural log

    RonL
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,899
    Thanks
    327
    Awards
    1
    Quote Originally Posted by Jameson
    In American schools we learn log to mean base 10 and ln to mean base e. For what reason we do this I don't know. Most higher level math literature I've read always uses log to mean base of e.
    Thank you for mentioning that, Jameson, because whenever I read "log" I automatically think log_{10} and it drives me crazy. (Well, crazier than usual. ) I almost always have to rewrite the expression so I can concentrate on the equations!

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. alternating series help
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 29th 2011, 04:28 PM
  2. alternating series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 26th 2010, 10:37 PM
  3. Alternating series
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 31st 2010, 10:44 PM
  4. alternating series
    Posted in the Math Challenge Problems Forum
    Replies: 1
    Last Post: August 2nd 2009, 04:17 AM
  5. Alternating Series
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 26th 2009, 03:37 AM

Search Tags


/mathhelpforum @mathhelpforum