Does the series (1+sin(n))/(10^n) from summation 0 to positive infinity converge or diverge?
I let an = (1+sin(n))/(10^n) and bn = 1/10^n
lim as n approaches positive infinity = an/bn = ((1+sin(n))/(10^n))/(1/(10^n))= 1+sin(n)= positive infinity.
I don't know if it's right or not but if someone could look over it that would be great.
I ran this simple Quick-Basic program
FOR N = 1 TO 20
TERM = (1+ SIN(N))/(10^N)
SUM = SUM + TERM
PRINT TERM, SUM
NEXT N
the terms keep getting smaller by appr a factor of 10, (obviously the denominators become 1,10,100,1000, ...)
since the sin(N) can only be a number between -1 and 1, then (1+sin(n)) will range between 0 and 2
after about 7 terms, the sum had approached a value of 1.204407 and since each successive term would be smaller than 10^-7, it would converge to that sum.
To determine whether the series converges or diverges, we need to consider the behavior of the terms as n approaches infinity. Your approach is on the right track, but there are a few errors in your calculation.
Let's recheck the calculation:
First, we have the series an = (1 + sin(n))/(10^n) and bn = 1/10^n.
Next, we can compare the behavior of an and bn to check if an converges or diverges. We can compare them by taking the limit as n approaches infinity of the quotient an/bn:
lim as n approaches infinity of (an/bn) = lim as n approaches infinity of (((1 + sin(n))/10^n)/(1/10^n))
= lim as n approaches infinity of (1 + sin(n))
Here's where the error lies:
The limit of (1 + sin(n)) as n approaches infinity does not exist, as the sine function oscillates between -1 and 1 indefinitely. Therefore, it is not valid to conclude that the limit is positive infinity based on that expression.
To properly evaluate this series, we need to take an alternative approach:
Since the limit of (1 + sin(n)) as n approaches infinity is undefined, we cannot directly compare an with bn using the limit quotient test. Instead, let's consider the behavior of an in the long run.
Notice that an oscillates between values close to 0 (when sin(n) = -1) and values close to 2 (when sin(n) = 1). Therefore, the series does not converge. It diverges since the terms do not approach zero.
In conclusion, the series (1 + sin(n))/(10^n) from n = 0 to positive infinity diverges.