Is the Series Convergent or Divergent?

[tnutty]

Homework Statement

Determine whether the series converges or diverges.

$$\sum$$ 3+7ⁿ / 6ⁿ

Attempt :

Comparison test :

3+7ⁿ / 6ⁿ < 7ⁿ / 6ⁿ

3+7ⁿ / 6ⁿ < (6/7)ⁿ

since (6/7)ⁿ is a geometric series and is convergent is
3+7ⁿ / 6ⁿ convergent as well?

 

[tnutty]

I see my mistake

3+7n / 6n > (7/6)^n

but then what?

 

[yyat]

Hint: For a series
$$\sum_{n=0}^{\infty}a_n$$

to converge, the terms have to converge to zero, i.e.

$$\lim_{n->\infty}a_n=0$$.

 

[tnutty]

I could use the limit comparison test but I get stuck.

 

[yyat]

You have $$a_n=3+7^n/6^n>3$$, so can $$a_n$$ converge to zero?

 

[tnutty]

How did you figure that inequality ?

(7/6)^n converges to -7

 

[yyat]

How did you figure that inequality ?

Clearly, (7/6)^n is a positive number for any n.

(7/6)^n converges to -7

I don't think so, 7/6>1, so (7/6)^n goes to infinity for large n.

 

[tnutty]

(7/6)^n

This is a geometric series. And I know that by definition a*r^(n-1) ,where |r|<1 = a/(1-r).

so (7/6)^n
=

(7/6) * (7/6)^(n-1)

so,
a = 7/6
r = 7/6

it follows that

(7/6)^n = a/(1-r) = (7/6) / (1-7/6) = -7

 

[tnutty]

wait I see what your saying. How comes this the above statement is wrong?

 

[tnutty]

no your right, 7/6 > 1 so this series diverges.
Thanks