You will need to check this carefully!
Last part:
Now:
So:
So now you need only find such that:
Which suggests that will suffice
RonL
"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.
Default meaning of log is natural log, a text will make it clear thatOriginally Posted by AfterShock
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
Impressive!!
Thank you for mentioning that, Jameson, because whenever I read "log" I automatically think and it drives me crazy. (Well, crazier than usual. ) I almost always have to rewrite the expression so I can concentrate on the equations!Originally Posted by Jameson
-Dan