Should Infinite Series Always Start from n=0 in Comparison Tests?

[neelakash]

Homework Statement

It is not a homework problem.I just want to clarify whether during the comparison tests of infinite series,should we start the series from n=0 whenever possible?

Homework Equations

The Attempt at a Solution

Actually,there are many series for which n=0 term is not defined and hence,it is not taken into account.But I saw for (1/2^n) geometric series,it is quite helpful to consider the zeroth term and hence to prove the convergence of (1/n^2) by appropriate grouping method.

So,we can use n=0 term of a series even when comparing it with a series starting with n=1 term.Right?

 

[anathema]

it is important to always follow a systematic approach when analyzing and comparing series. In general, it is best to start the series from n=1 and include all terms in the comparison. However, there may be cases where starting from n=0 or excluding certain terms may be beneficial, as you have observed with the (1/n^2) series. Ultimately, the approach may vary depending on the specific series and the techniques being used. It is always important to clearly state and justify the approach taken in your analysis.

 

[m2003]

I can provide a general response to this question.

In general, when using comparison tests for infinite series, it is not always necessary to start the series from n=0. The choice of starting point depends on the specific series being considered and the method being used for comparison. In some cases, starting from n=0 may be helpful in proving convergence, as seen in the example of the (1/2^n) geometric series. However, in other cases, the zeroth term may not be defined or may not provide relevant information for the comparison. It is important to carefully consider the specific series and choose the starting point that best suits the situation.