Determine whether the given series converges or diverges, and find the sum if it converges. I don't understand, help!! Thnx!
1) 20+5+(5/4)+(5/16)+...
this is just a geometric series, where
a = 20
r = 1/4
So, the sum is
S = a/(1-r) = 20/(3/4) = 80/3
To determine whether the given series converges or diverges, we need to analyze the behavior of the terms in the series.
The given series is: 20 + 5 + (5/4) + (5/16) + ...
The terms of the series form a geometric sequence with a common ratio of 1/4. A geometric sequence has the general form a, ar, ar², ar³, ... where 'a' is the first term and 'r' is the common ratio.
In this case, the first term 'a' is 20, and the common ratio 'r' is 1/4. Using this information, we can write the series as:
20 + 20(1/4) + 20(1/4)² + 20(1/4)³ + ...
To determine whether this series converges or diverges, we need to find the sum of the series. For a geometric series, the sum can be found using the formula:
S = a / (1 - r)
where S is the sum, 'a' is the first term, and 'r' is the common ratio.
Plugging in the values, we have:
S = 20 / (1 - 1/4)
Simplifying:
S = 20 / (3/4)
To divide by a fraction, we can multiply by its reciprocal:
S = 20 * (4/3)
Simplifying further:
S = 80/3
Therefore, the given series converges, and the sum is 80/3.