- Find using $ S_n = \sum_{n=1}^{\infty} \dfrac{8}{(-3)^{n}} $:
This problem aims to find the partial sum of a series where $n$ represents the number of outcomes. For better understanding, you should be familiar with the partial series formula and some basic graphing techniques.
A partial sum of a finite series can be defined as the summation of a limited number of successive values beginning with the first least value. If we encounter performing a partial sum with infinite series, it is usually valuable to analyze the behavior of partial sums.
Expert Answer
We will be working with geometric series, which is a series where the subsequent terms have a joint ratio. For instance, $1, 4, 16, 64$, …is known as an arithmetic sequence. A series constructed by utilizing a geometric sequence is known as the geometric series for instance $1 + 4 + 16 + 64$ …makes a geometric series.
The formula for a finite series is given by:
\[ s_n = \dfrac{a \left( 1-r^n \right)}{1-r} \hspace {3em} for \hspace {1em} r \neq 1, \]
Where,
$a$ is the first term,
$r$ is the common ratio and,
$s_n$ equals to $a_n$ for $r = 1$
We are given the following sum of series:
\[ s_n = \sum_{n=1}^{\infty} \dfrac{8}{(-3)^{n}} \]
When $n = 1$
\[ s_1 = \dfrac{8}{(-3)^1} = \dfrac{-8}{3} = -2.66667 \]
When $n = 2$
\[s_2 = \dfrac{8}{(-3)^1} + \dfrac{8}{(-3)^2} = \dfrac{-8}{3} + \dfrac{8}{9} = \dfrac{-16}{9} = -1.77778 \]
When $n = 3$
\[ s_3 = s_2 + \dfrac{8}{(-3)^3} = \dfrac{-16}{9} – \dfrac{8}{27} = \dfrac{-56}{27} = -2.07407 \]
When $n = 4$
\[ s_4 = s_3 + \dfrac{8}{(-3)^4} = \dfrac{-56}{27} + \dfrac{8}{81} = \dfrac{-160}{81} = -1.97531 \]
When $n = 5$
\[ s_5 = s_4 + \dfrac{8}{(-3)^5} = \dfrac{-160}{81} – \dfrac{8}{243} = \dfrac{-488}{243} = -2.00823 \]
When $n = 6$
\[ s_6 = s_5 + \dfrac{8}{(-3)^6} = \dfrac{-488}{243} + \dfrac{8}{729} = \dfrac{-1456}{729} = -1.99726 \]
When $n = 7$
\[ s_7 = s_6 + \dfrac{8}{(-3)^7} = \dfrac{-1456}{729} – \dfrac{8}{2187} = \dfrac{-4376}{2187} = -2.00091 \]
When $n = 8$
\[ s_8 = s_7 + \dfrac{8}{(-3)^8} = \dfrac{-4376}{2187} + \dfrac{8}{6561} = -1.99970 \]
When $n = 9$
\[ s_9 = s_8 + \dfrac{8}{(-3)^9} = -1.99970 – \dfrac{8}{19683} = -2.00010 \]
And finally, when $n = 10$
\[ s_10 = s_9 + \dfrac{8}{(-3)^10} = -2.00010 + \dfrac{8}{59049} = -1.99996 \]
Inserting the $10$ partial sums of the series in the table:
Figure 2
The graph of the filled table is given in blue, whereas the actual sequence is in red:
Figure 3
Numerical Result
The $10$ partial sums of the given series are $-2.66667$, $-1.77778$, $-2.07407$, $-1.97531$, $-2.00823$, $-1.99726$, $-2.00091$, $-1.99970$, $-2.00010$, $-1.99996$.
Example
Find $3$ partial sums of the series. $ \sum_{n=1}^{\infty} \dfrac{7^n + 1}{10^n} $
\[ n= 1, s_1 = \dfrac{7^2}{10} = 4.90 \]
\[ n= 2, s_2 = 4.90 + \dfrac{7^3}{10} = 8.33 \]
\[ n= 3, s_3 = 8.33 + \dfrac{7^4}{10} = 10.73 \]
The $3$ partial sums of the given series are $4.90$, $8.33$, $10.73$.