Which equation could be used to calculate the sum of

dictetzqh 2021-11-17 Answered
Which equation could be used to calculate the sum of the geometric series?
\(\displaystyle{\frac{{{1}}}{{{3}}}}+{\frac{{{2}}}{{{9}}}}+{\frac{{{4}}}{{{27}}}}+{\frac{{{8}}}{{{21}}}}+{\frac{{{16}}}{{{243}}}}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Thomas Conway
Answered 2021-11-18 Author has 405 answers

The formula for the nth partial sum of a geometric sequence is,
\(\displaystyle{S}_{{n}}={\frac{{{a}_{{1}}{\left({1}-{r}^{{n}}\right)}}}{{{1}-{r}}}},\ {r}\ne{1}\)
where, \(\displaystyle{S}_{{n}}\) is the sum of GP with n terms
\(\displaystyle{a}_{{1}}\) is the first term
r is the common ratio
n is the number of terms
The sum of the geometric series given is,
\(\displaystyle{\frac{{{1}}}{{{3}}}}+{\frac{{{2}}}{{{9}}}}+{\frac{{{4}}}{{{27}}}}+{\frac{{{8}}}{{{21}}}}+{\frac{{{16}}}{{{243}}}}\)
There are 5 terms in the series and the common ratio of the series is found by dividing any term by the previous term. Suppose, divide the second term by the first term and the common ratio is obtained as,
\(\displaystyle{r}={\frac{{{2}}}{{{9}}}}\div{\frac{{{1}}}{{{3}}}}\)
\(\displaystyle={\frac{{{2}}}{{{3}}}}\)
Therefore, r is not equal to 1 and the sum of the given series is,
\(S_5=\frac{\frac{1}{3}(1-(\frac{2}{3})^5)}{(1-\frac{2}{3})}\)

Have a similar question?
Ask An Expert
0
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-06-02
Find the value of x for which the series converges
\(\sum_{n=1}^\infty(x+2)^n\) Find the sum of the series for those values of x.
asked 2021-11-13
Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A circular swimming pool has a diameter of 24 ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? (Use the fact that water weighs 62.5lb/ft^3.)
asked 2021-11-08
Approximate the sum of the series correct to four decimal places.
\(\displaystyle{\sum_{{{n}}}^{\infty}}{n}={\frac{{{1}{\left(-{1}\right)}^{{n}}}}{{{3}^{{n}}{n}!}}}\)
asked 2021-11-23
Find whether the series diverges and its sum:
\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\left(-{1}\right)}^{{{n}+{1}}}{\frac{{{3}}}{{{5}^{{n}}}}}\)
asked 2021-11-13
Determine the sum of the following series.
\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\left({\frac{{{1}^{{n}}+{9}^{{n}}}}{{{11}^{{n}}}}}\right)}\)
asked 2021-10-17
A pair of honest dice is rolled once. Find the expected value of the sum of the two numbers rolled.
asked 2021-10-18
A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability that a 5 occurs first. Hint: Let
\(\displaystyle{E}_{{n}}\)
denote the event that a 5 occurs on the nth roll and no 5 or 7 occurs on the first n-1 rolls. Compute
\(\displaystyle{P}{\left({E}_{{n}}\right)}\)
and argue that
\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{P}{\left({E}_{{n}}\right)}\)
is the desired probability.

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question
...