Testing series for convergence

[DorianG]

Homework Statement

On a recent homework problem, I tested a series for convergence using the comparison test. If the first term of the series to be tested, a_n, was included, my test was inconclusive. From a problem that the lecturer did in class, he stated that the first few terms of an inifinte series don't matter when testing for convergence. So I compared the sum of the series from n=2 to infinity to my chosen series, showed it was smaller than this convergent series and this implied convergence of a_n.
The TA had my comment marked incorrect, stating that the first few terms can't be disregarded, so I'm unsure which is true.
Is it a case where, if you're calculating the sum, you can't disregard the first few terms, but if you're only testing for convergence, you can?
Thanks.

Homework Equations

The Attempt at a Solution

 

[CompuChip]

Let us assume that all the terms in the sum are finite (e.g. no $a_n$ is infinite). Then any finite sum
$$\sum_{n = 1}^k a_n$$
is finite. Since
$$\sum_{n = 1}^\infty a_n = \sum_{n = 1}^k a_n + \sum_{n = k + 1}^\infty a_n$$
there are two possibilities: the sum from k + 1 to infinity is finite, in which case the expression is (something finite + something finite = something finite), or it is infinite, and then it is (something finite + something infinite = infinite).
Of course, for calculating the value of the sum, you cannot neglect any terms.

There can be subtleties with non-absolutely converging sums and such, if you have any doubt I think the best thing to do is ask your TA why he marked it incorrect (e.g. if he can give you an example where it goes wrong) and/or go see the teacher.

 

[DorianG]

Thanks Compuchip, I appreciate the reply.