I have to get the result of the serie from 1 to infinite of
|sin(4n/π) +3|/ 4^n
I really don't know what to do...
Please I would really appreciate if anyone can help me
is the expression to be summed really |sin(4n/π)+3|/4^n,
or is it
|sin(4nπ)+3|/4^n?
It's |sin(4n/π)+3|/4^n
clearly, the series converges, since
|sin(4n/π)+3|/4^n < |1+3|/4^n
which converges to 16/3
But I can't come up with the actual limit value.
To find the sum of the series from 1 to infinity of |sin(4n/π) + 3| / 4^n, we can apply the concept of convergence and evaluate whether the series converges or diverges.
First, let's analyze the individual terms of the series. The term |sin(4n/π) + 3| / 4^n consists of the absolute value of the sine function with an argument of (4n/π) added to 3, divided by the exponential function 4^n.
Now, let's break down the steps to evaluate the convergence of this series:
Step 1: Simplify the expression within the absolute value.
Since the argument of the sine function ranges from -1 to 1, it is always between -4 and 4. Adding 3 to this value results in a range of -1 to 7. Therefore, the absolute value function has a range of 1 to 7.
Step 2: Analyze the numerator.
The numerator, |sin(4n/π) + 3|, has a minimum value of 1 and a maximum value of 7, as mentioned in the previous step.
Step 3: Analyze the denominator.
The denominator, 4^n, is an exponential function that grows with increasing values of n.
Step 4: Apply the limit comparison test.
To evaluate the convergence, we can apply the limit comparison test. We need to find a known series with terms that have the same growth pattern as our series.
Let's consider the series ∑ (1/4)^n, which is a geometric series with a common ratio of 1/4. This series converges because the common ratio is less than 1.
Step 5: Take the limit.
Now, we will calculate the limit of the ratio of the terms using L'Hôpital's rule. Let's denote the general term of our series as a_n.
lim(n→∞) a_n / (1/4)^n
We can rewrite a_n as |sin(4n/π) + 3| / 4^n.
lim(n→∞) {|sin(4n/π) + 3| / 4^n} / (1/4)^n
To apply L'Hôpital's rule, differentiate the numerator and denominator with respect to n:
lim(n→∞) [(d/dn) |sin(4n/π) + 3| / (d/dn) 4^n] / [(d/dn) (1/4)^n]
The derivative of |sin(4n/π) + 3| simplifies to (4/π)cos(4n/π). The derivative of 4^n is (4^n)ln(4), and the derivative of (1/4)^n is (-1/4)^n * ln(1/4).
lim(n→∞) [(4/π)cos(4n/π) / (4^n)ln(4)] / [(-1/4)^n * ln(1/4)]
Since lim(n→∞) cos(4n/π) oscillates between -1 and 1, we can ignore that term. Additionally, ln(1/4) is a constant, so we can remove the denominator. By simplifying further, we get:
lim(n→∞) (4/π) / (4^n) = 4 / (π * 4^n)
Step 6: Compare the limit with the known convergent series.
The limit we found is 4 / (π * 4^n), which is a constant multiple of the series ∑ (1/4)^n. Since ∑ (1/4)^n converges, our series ∑ {|sin(4n/π) + 3| / 4^n} also converges.
Therefore, the series from 1 to infinity of |sin(4n/π) + 3| / 4^n converges.
Now that we have established the convergence of the series, determining the exact sum is a challenging task.