Evaluate the geometric Series or state that it diverges
lower bound k= 1 upper bound = infinity
E (sum symbol) E 5(-1/6)^(5k)
I know it converges but im not sure to what
To evaluate the geometric series or determine if it converges, we need to calculate the sum of the terms. In this case, you have a geometric series with the first term (a) equal to 5 and the common ratio (r) equal to -1/6.
The sum of a geometric series can be calculated 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:
a = 5
r = -1/6
S = 5 / (1 - (-1/6))
= 5 / (1 + 1/6)
= 5 / (7/6)
= 30/7
Therefore, the sum of the given geometric series is 30/7.
To evaluate the given geometric series, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
In this case, the first term (a) is 5 and the common ratio (r) is -1/6.
Now let's substitute the values into the formula:
S = 5 / (1 - (-1/6))
= 5 / (1 + 1/6)
= 5 / (7/6)
To simplify, we can multiply the numerator and denominator by 6:
S = (5 * 6) / 7
= 30 / 7
Therefore, the given geometric series converges to the value of 30/7.
a = 5(-1/6)^5
Note that (-1/6)^(5k) = ((-1/6)^5)^k, so
r = (-1/6)^5
S = a/(1-r) = (5(-1/6)^5)/(1-(-1/6)^5) = -5/7777