Find the sum of the following convergent series:
(a) Ei=1 1 / (3i - 2)(3i + 1)
I know that it equals:
= 1/4 + 1/28 + 1/70 + 1/130 + 1/208
a= 1/4, but I can't figure out what r =
1/4 * 1/7 = 1/28, but from 1/28 to 1/70 and so on it doesn't work anymore.
(a) there is no r. It's not a geometric series
Just the formula shows that the ratio changes from term to term
S1 = 1/4
S2 = 1/4 + 1/28 = 2/7
S3 = 2/7 + 1/70 = 21/70 = 3/10
Looks like Sn = n/(3n+1)
Using induction,
Sk + 1/(3(k+1)-2)(3(k+1)+1)
= Sk + 1/(3k+1)(3k+4)
= k/(3k+1) + 1/(3k+1)(3k+4)
= (k(3k+4)+1)/(3k+1)(3k+4)
= (3k^2+4k+1)/(3k+1)(3k+4)
= (3k+1)(k+1)/(3k+1)(3k+4)
= (k+1)/(3(k+1)+1)
So that's ok.
So, if Sn = n/(3n+1) = 1/3 - 1/3(3n+1)
S∞ = 1/3
Ok I see, I was just adding the same fractions on every line instead of just adding the answer from the previous. Thanks Steve!
To find the sum of the convergent series, we can use the formula for the sum of an infinite geometric series. In this case, we need to determine the common ratio, notated as "r", in order to apply the formula.
The given series is:
Ei=1 1 / ((3i - 2)(3i + 1))
Let's simplify the expression in the denominator:
(3i - 2)(3i + 1) = 9i^2 + 3i - 6i - 2 = 9i^2 - 3i - 2
Now let's rewrite the series using the simplified expression:
Ei=1 1 / (9i^2 - 3i - 2)
To determine the common ratio, we need to express the series in a form that allows us to see the relationship between each term. Let's rewrite the series by factoring out the common ratio:
Ei=1 1 / ((9i^2 - 3i - 2) / (9))
Simplifying this expression gives:
Ei=1 9 / (9i^2 - 3i - 2)
Now, in order to determine the common ratio "r", we need to express consecutive terms of the series in a way that shows the relationship between them. To do this, we can rewrite the series as follows:
Ei=1 (1/9) * (1 / (i^2 - (1/3)i - (2/9)))
Now, let's expand the series to show the first few terms:
(1/9) * (1 / (1 - (1/3) + (2/9))) --> (1/9) * (1 / (4/9)) = 9/4
(1/9) * (1 / (4 - (4/3) + (2/9))) --> (1/9) * (1 / (20/9)) = 9/20
(1/9) * (1 / (9 - (3/3) + (2/9))) --> (1/9) * (1 / (16/9)) = 9/16
(1/9) * (1 / (16 - (2/3) + (2/9))) --> (1/9) * (1 / (42/9)) = 9/42
As observed, the series is a harmonic series with a common ratio that is not constant. Therefore, the series does not meet the criteria of a convergent geometric series. Thus, we cannot find a sum using the formula for the sum of a geometric series.
In conclusion, the given series does not have a finite sum.