Question

Determine if each statement is True or False.

Answer 1

Answer:

1) False 2) True 3) True 4) True

Step-by-step explanation:

1)FALSE

We can prove this by giving a counterexample,

Take the arithmetic sequence  

$\left \{ a_1,a_2,a_3,... \right \}$

where  

$a_n=(n-1)-15$

in this case d=1

Then

$a_1+a_2+...+a_12=-15-14-...-4$

2)TRUE

Given that for an arithmetic sequence

$a_n=a_1+(n-1)d$

Where d is a constant other than 0, then

$\lim_{n \to \infty}a_n\neq 0$

and so, the series  

$\sum_{n=1}^{\infty}a_n$

diverges.

3)TRUE

This is the definition of infinite sum.

If $S_n=a_1+a_2+...+a_n$

then $\sum_{n=1}^{\infty}a_n=\lim_{n \to \infty}S_n$  

4)TRUE

If  

$\left \{ a_1,a_2,a_3,... \right \}$

is a geometric sequence, then the n-th partial sum is given by

$S_n=\frac{a_1r^n-a_1}{r-1}$

Since r<1

$\lim_{n \to\infty}r^n=0$

and so, the geometric series

$\sum_{n=1}^{\infty}a_n=\lim_{n \to\infty}S_n=\frac{a_1}{1-r}$