Endpoint Analysis - McLauren Series

[sunburnt]

Hello there.

A function f is defined by: f(x) = 1/3 + 2/3²x + 3/3³x² + ... + (n + 1) / (3^{n + 1}) * xⁿ

I have found the interval of convergence of this McLauren series to be 3 ( i.e. -3 < x < 3) however, I am a little rusty on endpoint analysis. Can anyone help me determine whether or not -3 and/or 3 should be included? I plugged -3 into the function and came up with

1/3 - 2/3 + 1 - 4/3 ...
The signs alternate, however the values don't decrease, so I'm hesitant to say that it converges by the alternating sign test. It fails the Nth term test...

When I put in 3 I got...
1/3 + 2/3 + 1 + 4/3 which obviously doesn't converge.

Heeelp!

[krtxmrtz]

For x = -3:

f(-3) = 1/3 - 2/3 + 3/3 - 4/3 + 5/3 - 6/3 + ... =
1/3 (1 - 2 + 3 - 4 + 5 - 6 + ...)

If now you add every 2 terms you can write this as

f(-3) = 1/3 (-1 -1 -1 -1 - ...) which clearly does not converge.

Another way to see this: notice it does not necessarily tend to minus infinity for if you keep adding each term to the previous one you have an alternance of sign for the ever increasing partial sums:

1
-1
2
-2
3
-3
4
-4
...
(neglecting the constant factor 1/3)

This should be sufficient to prove the non-convergence.

[sql_lall]

Umm... just for clarification, when you write fractions, can you put whatever is 'above the line' to the left of the "/", and whatever is below the line to the right??

I.e. you are saying that 2/3²x =2/3 for x = 3
however, some people read 2/3²x as 2/(3²x), which has a different value (in this case, 2/27)
It seems that the series is more likely to converge if the powers of x are divided, not multiplied.

[krtxmrtz]

Originally posted by sql_lall
...you are saying that 2/3²x =2/3 for x = 3
however, some people read 2/3²x as 2/(3²x), which has a different value (in this case, 2/27)

Yes that was ambiguous but the interpretation is clear from the last (n) term in the original post above.
On the other hand it's no wonder you came up with this suggestion, you being in Australia, everything must be looking upside down to you

[sunburnt]

I thought of that after I posted it, but I'm just lazy. Thanks for the help guys.

[sql_lall]

Yeah, everything looks upsidedown cos i've been looking at upside-down maps all my life. The REAL maps (peterson projection) show land area in Proper ratio, not having Nothern Hemisphere enlarged, AND the have south up the top, which, as we all know, looks MUCH nicer

[krtxmrtz]

Well, it must be certainly nicer cause you don't have to wear those heavy lead-loaded boots to keep you from falling away into outer space...