Question

Given the infinite series: 5%29%7D%2B%5Cfrac%7B1%7D%7B%285%29%287%29%7D%2B..." id="TexFormula1" title="\frac{1}{(1)(3)} +\frac{1}{(3)(5)}+\frac{1}{(5)(7)}+..." alt="\frac{1}{(1)(3)} +\frac{1}{(3)(5)}+\frac{1}{(5)(7)}+..." align="absmiddle" class="latex-formula">
a. Rewrite the series in sigma notation
b. Write an explicit formula for the nth partial sum

Answer 1

a) The infinite series in sigma notation is described by this expression:

$y = \sum \limits_{i=1}^{\infty} \frac{1}{i\cdot (i+2)}$     (1)

b) The explicit formula for the n-th partial sum is represented by the following expression:

$y = \frac{1}{i\cdot (i+2)}$, i ∈ $\mathbb{N}$     (2)

How to derive an expression for a monotonous series

An infinite series is monotonous when it is bounded, that is, when the limit of the infinite series exists. In this case, we have an evidence of monotony in the denominators of the terms of the given series. In two consecutive terms, the latter always have a denominator greater than the former.

a) The series in sigma notation is now described below:

$y = \sum \limits_{i=1}^{\infty} \frac{1}{i\cdot (i+2)}$

b) The explicit formula for the n-th partial sum is defined by the expression within the sum, which is now presented below:

$y = \frac{1}{i\cdot (i+2)}$, i ∈ $\mathbb{N}$  

To learn more on infinite series: brainly.com/question/4268280

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