# Convergence and divergence of series when changing finite number of summands

I found a text saying that adding or changing only a finite number of summands does not have an effect on the convergence/divergence of the series. This is shown by the following argumentation:

Let $$sum_{k=n_1}^{infty} a_k$$ and $$sum_{k=n_2}^{infty}b_k$$ be two series with $$(s_n)_{n geq n_1}$$ and $$(t_n)_{t geq n_2}$$ their partial sums. Let’s suppose there exists an $$N$$ so that $$a_k = b_k$$ for alle $$kgeq N$$, than we have

begin{align}s_n = sum_{k=n_1}^{n}a_k = a_{n_1} + a_{n_1+1} + ldots + a_{N-1} + sum_{k=N}^{n}a_kend{align} and

begin{align}t_n &= sum_{k=n_2}^{n}b_k = b_{n_2} + b_{n_2+1} + ldots + b_{N-1} + sum_{k=N}^{n}a_k \ &= s_n – left(a_{n_1} + a_{n_1+1} + ldots + a_{N-1}right) + left(b_{n_2} + b_{n_2+1} + ldots + b_{N-1}right)end{align}

for all $$n geq N$$. Hence, both $$(s_n)_{n geq n_1}$$ and $$(t_n)_ {tgeq n_2}$$ are either convergent or divergent.

Unfortunately I do not see why $$(s_n)_{n geq n_1}$$ and $$(t_n)_{t geq n_2}$$ are either convergent or divergent following this calculation. Moreover I also don’t get why this is showing that a finite number of changes to the summands of the series does not change the convergence behaviour of the series.
Can someone please help me understanding this proof.

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