Question

Someone please help, question about infinite series!

Analyze the infinite series below.

120, 24, 4.8, 0.96, …

Does the series have a sum? If so, what is the sum? If not, why not?

Answer 1

yes it has

Answer 2

its a geometric progression

Answer 3

An infinite series has a sum if the partial sums form a sequence that has
a real limit. So add up the first two, then add up the first three, then add up all four and see what number it comes close to. That will be the limit.

Answer 4

If no limit not a sum

Answer 5

no i dont think so youll have to do the tedious work @fmg78360 ans is 150

Answer 6

144 , 144 , 148.8, 149.76. So @Meinme is correct 150.

Answer 7

formula is a/1-r but only works for r<1

Answer 8

\[120+ 24+4.8+0.96+\cdots\]
If you were to check, you'd see that the common ratio between successive terms is 1/5 (120/5 = 24, 24/5 = 4.8, and so on).

As @Meinme suggested, the series will only converge if the ratio is bounded by 1, i.e. |r| < 1. So, since you're given a geometric series with a ratio that satisfies this condition. Therefore, the series converges and has a sum.
\[\sum_{n=0}^\infty120\left(\frac{1}{5}\right)^n\]
The sum of a series of the form on the LHS is the expression on the RHS:
\[\sum_{n=0}^\infty ar^n=\frac{a}{1-r}\]

Answer 9

Wait, so I'm confused as to what the sum is?

Answer 10

The last bit of my most recent post: That's the formula you use. That is the sum.

Answer 11

I'm still a bit confused... so the "a/(1-r)" part is the sum or what?

Answer 12

Using that formula, the sum for your particular series is\[\frac{120}{1-\frac{1}{5}}\]
because a = 120 and r = 1/5.

Answer 13

So 150 is the sum, right?

Answer 14

yup

Answer 15

Thanks a bunch!