Upper Estimate

$$ \sum_{n=1}^\infty a_n \le\,
a_1 + \int_1^\infty f(x)\, dx$$
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Lower Estimate

$$ \sum_{n=1}^\infty a_n \ge
\int_1^\infty f(x) \,dx$$
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Bounds on series

$$ \int_1^\infty f(x) \,dx \le
\sum_{n=1}^\infty a_n \le\, a_1 + \int_1^\infty f(x)\, dx$$

(after putting together the estimates from above)

Estimate of series

$$s_{n1}+\int_n^\infty
f(x)\,dx\le\sum_{i=1}^\infty a_i\le s_n+\int_n^\infty
f(x)\,dx$$

Explained in the video.
