Question

Determine whether the sequence converges or diverges. If it converges, give the limit. 60, -10, 5/3 , -5/18 , ...

Diverges

Converges; 11100

Converges; 72

Converges; 0

Thanks!

Answer 1

Intuitively: Each successive number is getting smaller (in absolute value), and the smallest it can get is 0.

More explicitly, we can see that the
`n`-th term in the sequence is given by
`60\cdot6^(1-n)`. As
`n` gets arbitrarily large,
`6^(1-n)` gets closer and closer to 0.

Answer 2

Explanation:

Given the sequence

60, -10, 5/3, -5/18

Let the general formula for the sequence

You will notice that the sequence is has a fluctuating sign, from positive to negative and it has a common ratio of -1 / 6

Then, it is a GP.

So, the n-th term of a GP is determine by

An = ar^(n-1)

Where a is first term a = 60

r is common ration r = -1/6

An is the n-th term

Then,

An = 60 × (-1/6)^(n-1)

Using Limit

Note that as n→∞

(-1/6)^(n-1) → (-1/6)^∞ = 0

An → 60 × 0 = 0

So the limit converges since the limit is zero.

Now, we can also use the formula of GP which is the sum to infinity of a series to find where it converges to.

S∞ = a / (1-r). .,..0< r<1

S∞ = 60 / 1 - (-1/6)

S∞ = 60 / (1 + 1/6)

S∞ = 60 / 7 / 6

S∞ = 60 × 6 / 7

S∞ = 51.43

So, the series converges to 51.43