Consider the sequence of sums 1,1+2,1+2+3, 1+2+3+4,1+2+3+4+5, …. (a) What happens to the terms o

step 1 So we want to consider this sequence here where it's just one, one plus two, one plus two plus t

Consider the sequence of sums 1,1+2,1+2+3,...

Consider the sequence of sums $1,1+2,1+2+3$, $1+2+3+4,1+2+3+4+5, \ldots .$
(a) What happens to the terms of this sequence of sums as $k$ gets larger and larger?
(b) Find a sufficiently large value of $k$ that will guarantee that every term past the $k$ th term of this sequence of

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Key Concepts

Divergence

Divergence refers to the behavior of a sequence where its terms increase without bound as the index grows. Understanding divergence is important because it informs us that for sequences like the sum of natural numbers, there is no finite limit, and the terms eventually exceed any given bound.

Closed-Form Expression

A closed-form expression is a formula that directly expresses the nth term of a sequence as a function of n, without the need for recursion or iteration. In the context of summing natural numbers, the closed-form expression n(n+1)/2 provides an efficient method to determine the sum up to any given term.

Sequence of Partial Sums

A sequence of partial sums is formed by summing a sequence's elements consecutively, where each term in the new sequence represents the total of all preceding terms in the original sequence. This concept is essential for analyzing cumulative behavior in sequences.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term increases by a constant difference. This type of series has a straightforward closed-form formula, enabling quick calculation of any term in the sequence without performing all intermediate additions.

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