Let an ≥0 for all n, and let sn=∑i=1^n ai. Show that (sn) converges if and only if (sn) is bound

step 1 Hello. So recall, a sequence S sub N is set to be a Koshi sequence if and only if. For every eps

Let an ≥0 for all n, and let sn=∑i=1^n ai. Show that (sn)...

Let $a_n \geq 0$ for all $n$, and let $s_n=\sum_{i=1}^n a_i$. Show that $\left(s_n\right)$ converges if and only if $\left(s_n\right)$ is bounded. Recall that a sequence of real numbers $\left(x_n\right)$ is said to be Cauchy if, for every $\varepsilon>0$, there is an integer $N \geq 1$ such that $\left|x_n-x_m\right|<\varepsilon$ whenever $n, m \geq N$.

Let $a_n \geq 0$ for all $n$, and let $s_n=\sum_{i=1}^n a_i$. Show that $\left(s_n\right)$ converges if and only if $\left(s_n\right)$ is bounded.

Recall that a sequence of real numbers $\left(x_n\right)$ is said to be Cauchy if, for every $\varepsilon>0$, there is an integer $N \geq 1$ such that $\left|x_n-x_m\right|<\varepsilon$ whenever $n, m \geq N$.

Key Concepts

Monotone Sequences

A monotone sequence is one that is either non-decreasing or non-increasing throughout. When dealing with series whose terms are non-negative, the sequence of partial sums becomes monotone increasing. This property simplifies the analysis of convergence because monotone sequences have the special characteristic that boundedness implies convergence.

Bounded Sequences

A bounded sequence is one for which there exists a real number M such that every term of the sequence is less than or equal to M in absolute value. In the setting of a monotone increasing sequence, such as the partial sums of non-negative terms, being bounded ensures that the sequence does not diverge to infinity, which is a key step in proving convergence.

Monotone Convergence Theorem

The Monotone Convergence Theorem states that every monotone (increasing or decreasing) sequence that is bounded converges to a limit. This theorem is fundamental when analyzing series with non-negative terms since it allows one to conclude that if the partial sums are bounded, the series converges.

Cauchy Sequences

A Cauchy sequence is defined by the property that for every positive epsilon, there exists an index beyond which the distance between any two terms is less than epsilon. In complete metric spaces like the real numbers, every Cauchy sequence converges. This concept is crucial because it provides an alternative definition of convergence that is independent of the limit and is especially useful in abstract spaces.

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