Question

Determine if each statement is True or False.

1. Suppose an is an arithmetic sequence with d > 0. Then the sum
of the series a1+a2+a3+...a12 must be positive.
2. All infinite arithmetic series diverge.
3. An infinite sum is equal to the limit of the sequence of partial
sums.
4. An infinite geometric series will converge if r < 1.

Answer 1

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

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<0`

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\\eq 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=(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=(a_1)/(1-r)`