Say I have to prove that $\sum_{n=0}^\infty \frac{1}{\sqrt{n}} $ is divergent. $\sum_{n=0}^\infty \frac{1}{\sqrt{n}} $ is just $\sum_{n=1}^\infty \frac{1}{\sqrt{n}} $ plus an additional term. Since $\sum_{n=1}^\infty \frac{1}{\sqrt{n}} $ is divergent, adding a finite number of terms doesn't do anything to the divergence of the series so $\sum_{n=0}^\infty \frac{1}{\sqrt{n}} $ is divergent. What I'm wondering is if this work is necessary? Does the p-series test only apply to infinite series starting at $n=1$ or can the starting $n$ value be anything? If the latter is the case, then my proof is uneccesary and I could just cite the p-series test and be done. What about for all the other series tests? Do they always require that $n$ start at $1$?
Secondly, say I have to prove that $\sum_{n=2}^\infty \frac{1}{\sqrt{n}} $ is divergent. Could I say that because $\sum_{n=1}^\infty \frac{1}{\sqrt{n}} $ is divergent, subtracting a finite number of terms from the sum doesn't change the fact that the series is divergent and thus $\sum_{n=2}^\infty \frac{1}{\sqrt{n}} $ is divergent?
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$\begingroup$In general, yes, adding or subtracting (or changing in any way) a finite number of terms does not change the series behaviour; $\sum_{n=1}^{\infty}a_n$ converges if and only if $\sum_{n=k}^{\infty}a_n$ does.
However, this assumes all the terms in question actually exist - your original question about $\sum_{n=0}^\infty\frac1{\sqrt n}$ doesn't make sense, because the first term of the sum is $1/0$, which is undefined. So none of the partial sums exist and they therefore can't be said to converge (or diverge).
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