∑from1 to ∞((6^n +5)/(6^n+1)
as n gets huge, the terms approach 6^n/6^n = 1 (6^n is so huge that the +5 and +1 make no difference)
for any series to converge, the terms of the sequence need to go to zero.
These terms approach 1, so the series becomes
...+ 1+1+1+1...
no way that can converge
you can also try the ratio test. The ratio of two successive terms must be less than 1. Yours is exactly 1.
lim(n->∞) (6^n +5)/(6^n+1) = 1
so I doubt the series converges
Are you using the diverges test? How did you got 1?
To find the sum of the series ∑(6^n + 5)/(6^(n+1)) from n = 1 to ∞, we can start by simplifying the terms of the series.
The term of the series can be written as:
(6^n + 5)/(6^(n+1))
Now, let's simplify the term by expanding the power of 6 in the denominator:
(6^n + 5)/(6 * 6^n)
Now, we can simplify further by canceling out a common factor of 6^n from the numerator and the denominator:
(1 + 5/6^n)
Now, let's rewrite the series with the simplified term:
∑(1 + 5/6^n)
We have separated the series into two parts, where the first part is the sum of 1 and the second part is the sum of 5/6^n.
The sum of the first part (1) is simply 1, as it is a constant value.
To find the sum of the second part (5/6^n), we can use the formula for the sum of a geometric series. The sum of a geometric series is given by:
S = a / (1 - r)
In our case, the first term (a) is 5/6^1 = 5/6, and the common ratio (r) is 1/6.
Substituting these values into the formula, we can find the sum of the second part:
S = (5/6) / (1 - 1/6)
= (5/6) / (5/6)
= 1
Therefore, the sum of the second part (5/6^n) is 1.
Now, we add the sum of the first part (1) and the sum of the second part (1) to find the overall sum of the series:
1 + 1 = 2
So, the sum of the series ∑(6^n + 5)/(6^(n+1)) from n = 1 to ∞ is equal to 2.