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
[tex]\left \{ a_1,a_2,a_3,... \right \}[/tex]
where
[tex]a_n=(n-1)-15[/tex]
in this case d=1
Then
[tex]a_1+a_2+...+a_12=-15-14-...-4<0[/tex]
2)TRUE
Given that for an arithmetic sequence
[tex]a_n=a_1+(n-1)d[/tex]
Where d is a constant other than 0, then
[tex]\lim_{n \to \infty}a_n\neq 0 [/tex]
and so, the series
[tex]\sum_{n=1}^{\infty}a_n[/tex]
diverges.
3)TRUE
This is the definition of infinite sum.
If [tex]S_n=a_1+a_2+...+a_n[/tex]
then [tex]\sum_{n=1}^{\infty}a_n=\lim_{n \to \infty}S_n[/tex]
4)TRUE
If
[tex]\left \{ a_1,a_2,a_3,... \right \}[/tex]
is a geometric sequence, then the n-th partial sum is given by
[tex]S_n=\frac{a_1r^n-a_1}{r-1}[/tex]
Since r<1
[tex]\lim_{n \to\infty}r^n=0[/tex]
and so, the geometric series
[tex]\sum_{n=1}^{\infty}a_n=\lim_{n \to\infty}S_n=\frac{a_1}{1-r}[/tex]
