Proof of Convergence/Divergence Unaffected by Series Starting Point

[adamg]

if we know that an infinite series is convergent from an integer T, to infinity, then the series is convergent from 1 to infinity. conversely, if a series is convergent from 1 to infinity then it is convergent from T to infinity (i.e. starting point of the series does not affect convergence/divergence) This seems obvious but can anyone help me prove it please.

 

[Galileo]

This is obvious from the definition of a convergent series. So check the definition again.

 

[Zurtex]

I pulled my lecturer up on this recently, consider the series:

$$\sum_{x=10}^{\infty} \frac{1}{(x-7)^2}$$

It converges and if you are interested to:

$$\frac{1}{12} \left(2 \pi^2 - 15\right)$$

However:

$$\sum_{x=1}^{\infty} \frac{1}{(x-7)^2}$$

Clearly does not converge, so be careful how you word it. Anyway, it's not too difficult to prove, just think of it like:

$$a_1 + a_2 + \ldots + a_{t-1} + \sum_{n=t}^{\infty} a_n$$

 

[mathman]

Is the series adamg asking about a power series? If so, the answer to his question is yes. However in the more general case as Zurtex showed, it is not true.

 

[Galileo]

I'd rather say his example

$$\sum_{x=1}^{\infty} \frac{1}{(x-7)^2}$$

is not a series, since the 7'th term is not defined.