we know the series from n=0 to infinity of c(sub n)*3^n converges
1. TRUE OR FALSE
the series from n=0 to infinity of c(sub n)*2^n must converge
2. TRUE OR FALSE
The series from n=0 to infinity of c(sub n)*4^n must converge
3. TRUE OR FALSE
the series from n=0 to infinity of c(sub n) must converge
MY ANSWERS I WANT CHECKED
1-true
2-false
3-true
To determine whether these statements are true or false, we need to use the ratio test. The ratio test helps us analyze the convergence of a series by comparing the absolute value of each term with the next term.
To apply the ratio test, let's consider the general series from n=0 to infinity of c(sub n)*r^n, where c(sub n) and r are positive numbers.
1. The series from n=0 to infinity of c(sub n)*2^n:
We compare the terms c(sub n)*2^n and c(sub n+1)*2^(n+1):
|c(sub n+1)*2^(n+1)| / |c(sub n)*2^n| = (c(sub n+1)/c(sub n)) * (2/2) = (c(sub n+1)/c(sub n))
Since we know that the series from n=0 to infinity of c(sub n)*3^n converges, it implies that the ratio (c(sub n+1)/c(sub n)) approaches a finite value as n approaches infinity. Let's denote this finite value as L.
If 0 < L < ∞, the series converges. If L = 0 or L = ∞, the series diverges.
Therefore, for the series from n=0 to infinity of c(sub n)*2^n, we can say:
- If L = 0, the series converges. In this case, the statement "the series from n=0 to infinity of c(sub n)*2^n must converge" is TRUE.
2. The series from n=0 to infinity of c(sub n)*4^n:
Similarly, we compare the terms c(sub n)*4^n and c(sub n+1)*4^(n+1):
|c(sub n+1)*4^(n+1)| / |c(sub n)*4^n| = (c(sub n+1)/c(sub n)) * (4/4) = (c(sub n+1)/c(sub n))
Using the same reasoning as before, we find that the ratio (c(sub n+1)/c(sub n)) approaches a finite value L as n approaches infinity.
- If 0 < L < ∞, the series converges. In this case, the statement "The series from n=0 to infinity of c(sub n)*4^n must converge" is FALSE.
3. The series from n=0 to infinity of c(sub n):
In this case, we don't have any powers of r, so no comparison is needed. We can directly determine whether the series converges based on the values of c(sub n) themselves.
Since the series from n=0 to infinity of c(sub n)*3^n converges, it means that the terms c(sub n) approach zero as n approaches infinity.
- If c(sub n) approaches zero as n approaches infinity, then the series converges. In this case, the statement "the series from n=0 to infinity of c(sub n) must converge" is TRUE.
Therefore, reviewing your answers:
1. TRUE (Correct)
2. FALSE (Correct)
3. TRUE (Correct)