Note: Enter your answer and show all the steps you use to solve this problem in the provided space.
How do the graphs of y=1/x and y = 5/x+6 compare?
To compare the graphs of y=1/x and y = 5/x+6, we can analyze their basic characteristics:
1. As x approaches infinity, both y=1/x and y=5/x+6 approach zero (they both have a horizontal asymptote at y=0).
2. As x approaches negative infinity, both y=1/x and y=5/x+6 approach zero, but y=5/x+6 is shifted up by 6 units (it has a horizontal asymptote at y=6).
3. At x=0, y=1/x is undefined, while y=5/x+6 is equal to 6.
4. Both functions have a vertical asymptote at x=0.
Using this information, we can sketch the graphs of the two functions and compare them:
(Note: These graphs are not to scale and are only intended to show a general comparison)
y=1/x (in blue) and y=5/x+6 (in red):
As shown in the graph, y=5/x+6 is a vertical shift of y=1/x by 6 units, meaning that its graph is the same as y=1/x but moved up by 6 units. Therefore, y=5/x+6 will always be higher than y=1/x for any given x value.
Evaluate the series 8∑ n=3 5n
125
38
210
165
We need to plug in each value of n from 3 to 8 into the expression 5n, then add up all the resulting terms, and finally divide by 125. So we have:
8∑n=3 5n/125 = (5×3 + 5×4 + 5×5 + 5×6 + 5×7 + 5×8)/125
= (15 + 20 + 25 + 30 + 35 + 40)/125
= 165/125
= 1.32
Therefore, the value of the series is 1.32 (rounded to two decimal places).
Answer: 165
what is the sum of the geometric series 10∑n=1 6(2) ^n
15,658
6,138
12,276
756
The sum of a geometric series can be found using the formula:
S = a(1 - r^n) / (1 - r)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 6(2)^1 = 12 and r = 2. Also, the number of terms is 10.
So we can plug these values into the formula and simplify:
S = 12(1 - 2^10) / (1 - 2)
= 12(1 - 1024) / (-1)
= 12(1023)
= 12,276
Therefore, the sum of the geometric series is 12,276.
Answer: 12,276
Does the infinite geometric series diverge or converge? Explain.
1/5 + 1/10 + 1/20 + 1/40
it diverges; it has a sum
it diverges; it does not have a sum
it converges ; it has a sum
it converges; it does not have a sum
This is a geometric series with first term a = 1/5 and common ratio r = 1/2. We can write the series as:
1/5 + 1/10 + 1/20 + 1/40 + ...
To determine if the series converges or diverges, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
If the value of S exists (i.e. it is a finite number), the series converges. If it does not exist (i.e. it is infinite or undefined), the series diverges.
Applying this formula to the given series, we get:
S = (1/5) / (1 - 1/2) = (1/5) / (1/2) = 2/5
Since the value of the sum S exists and is finite, the series converges.
Answer: It converges; it has a sum.