Can I use the index of a series for help with divergence? I was studying this series:
$$\sum_{n=2}^{\infty}\dfrac{5}{7n+28}$$
I know that it's an increasing, monotone sequence.
Also, I know I can rewrite as:
$$\sum_{n=2}^{\infty}\dfrac{5}{7(n+4)} = \dfrac{5}{7} \cdot \sum_{n=2}^{\infty}\dfrac{1}{n+4}$$
Also, I know that the following series is called the Harmonic Series and can be shown to diverge by over-estimating grouped terms (essentially at a point it is worse than adding $1/2$ over and over).  (Also, by Cauchy Condensation):
$$\sum_{n=1}^{\infty}\dfrac{1}{n}$$
I tried to setup a comparison test, but I don't think it works since the Harmonic is always larger:
$$0 \lt \dfrac{1}{n+4} \lt \dfrac{1}{n}$$
Is it legitimate to massage the index without affecting divergence/convergence?
For example,
The first few terms in the sequence of the series:
$$\sum_{n=2}^{\infty}\dfrac{1}{n+4}$$
Are $$\dfrac{1}{6},\dfrac{1}{7},\dfrac{1}{8},\dfrac{1}{9},...$$
Which looks just like the Harmonic just starting off at a different place.
So can I somehow change
$$\sum_{n=2}^{\infty}\dfrac{1}{n+4}$$  to $$\sum_{n=-3}^{\infty}\dfrac{1}{n+4}$$
to show it diverges?  Or am I just way off track here?

Notice that by changing the index we have
$$\sum_{n=2}^\infty\frac1{n+4}=\sum_{n=6}^\infty\frac1n$$
so the series is divergent. Notice also that the nature of a series doesn't depend on the first few terms which means that the two series $\sum\limits_{n\ge1}u_n$ and $\sum\limits_{n\ge n_0}u_n$ (for any $n_0$) have the same nature (both convergent or both divergent).
