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I just have a few quick questions on using the root test to determine the convergence a series. If I get a series $\sum_{n=5}^\infty A_n$. What would the $n=5$ do? From my knowledge when I apply root test, I use the limit as $n\to\infty$ anyways and ignore the $n=5$.

Finally if I get a series such as $\left(\frac{2\ln(n)}{\ln(n)+|\sin(n)|}\right)^n$ and I use the root test to test for the limit as n-> infinity, what if part of the series -- $\lim_{n\to\infty} |\sin(n)| = DNE$? Would it make my whole series divergent or can I just choose to ignore it as it only takes on values $0$ to $1$ and $2\ln(n)/\ln(n)$ is far greater as $n\to\infty$?

Sorry if my questions are unclear, it's my first time posting here and I'm quite confused in the first place so it's hard for me to explain.

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up vote 1 down vote accepted

The series $\displaystyle\sum_{n\ge0} a_n$ is convergent if and only if the partial sum sequence $ \left(\displaystyle\sum_{k=0}^n a_k\right)_n$ is convergent which's equivalent to the convergence of the sequence $ \left(\displaystyle\sum_{k=n_0}^n a_k\right)_n$ for all $n_0\in\Bbb N$ hence the convergence of a series doesn't depend of its first few terms.

For your second question notice that a series $\displaystyle\sum_{n\ge0} a_n$ is divergent when the sequence $(a_n)$ doesn't converge to $0$.

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