# 1. Find the first three terms of the sequence a_n= \frac{(-6)^{sin ((2n+1)^n/2)}}{(n+1)!} 2....

## Question:

1. Find the first three terms of the sequence

{eq}a_n= \frac{(-6)^{sin ((2n+1)^n/2)}}{(n+1)!} {/eq}

2. Find the first three terms of the sequence {eq}a_n= \frac{n!}{(n+4)!} {/eq}

3. {eq}5+.0005+.0000005 +...= {/eq}

4. {eq}\sum_{n=0}^{\infty} \frac{1}{(10n-7)- \frac{1}{(10n+3)}} {/eq}

5. {eq}\sum_{n=0}^{\infty} \frac{-10}{(100n^2-25)}{/eq}

## Definition of Sequence, Geometric Series and Comparison Test:

Let {eq}\displaystyle f:\mathbb{N}\to \mathbb{R} {/eq} be a real valued function, where {eq}\displaystyle \mathbb{N} {/eq} denotes the set of all natural numbers, i.e. the set of all positive integer and {eq}\displaystyle \mathbb{R} {/eq} is set of all real numbers. Then {eq}\displaystyle \{f(n)\}=\{f(1),\;f(2), \;f(3), \cdots \} {/eq} is called a real infinite sequence or simply a sequence. Usually a sequence denoted by {eq}\displaystyle \left\{ {{a_n}} \right\} {/eq}, where {eq}{a_n} = f\left( n \right) ,\; n\in \mathbb{N} {/eq}.

If we have an infinite series in the form {eq}\displaystyle k+kr+kr^2+kr^3+kr^4+ \cdots kr^n {/eq}, where k is a constant and r is a common ratio between terms,

then we have an infinite geometric series. When {eq}|r| < 1 {/eq}, the sum of such a series is given by {eq}\displaystyle \frac{k}{1-r} {/eq}.

Sometimes, a series given in summation notation won't be automatically recognizable as a geometric series,

but we can rewrite it using properties of series to take advantage of the geometric series formula.

If {eq}\sum\limits_{u = 1}^\infty {{w_k}} {/eq} and {eq}\sum\limits_{u = 1}^\infty {{c_k}} {/eq} be nonnegative series and

if {eq}\sum\limits_{u = 1}^\infty {{c_k}} {/eq} converges and {eq}0 \leqslant {w_k} \leqslant {c_k} {/eq} for all {eq}k \geqslant u {/eq} ,then {eq}\sum\limits_{l = 1}^\infty {{w_k}} {/eq} converges.

if {eq}\sum\limits_{l = 1}^\infty {{c_k}} {/eq} diverges and {eq}0 \leqslant {c_k} \leqslant {w_k} {/eq} for all {eq}k \geqslant 1 {/eq} ,then {eq}\sum\limits_{l = 1}^\infty {{w_k}} {/eq} diverges.