Question

Calculate the limit of the following sequences and find an ( n_{0} threshold for varepsilon>0 .

1. 1 √n

2. (2 n+1) (n+1)

3. (5 n-1) (7 n+2)

4. 1/( n−√n )

5. (1+…+n)/n²

6. ( √1 +√2 +…+√n )/n ⁴/³

7. n⋅√(1+(1/n) −1)

8. √n² +1+√n² −1 −2n

Answer 1

The provided question asks for the limits of several sequences as n approaches infinity and the determination of an n_0 threshold for a given epsilon. The limits vary and include 0, 2, 5/7, 1/2 and some require an application of the binomial theorem or other mathematical properties for finding the limit.

Step-by-step explanation:

The question asked is about determining the limits of several sequences as n approaches infinity and finding a threshold n_0 for a given epsilon (ϵ). To find these limits, we can apply various mathematical tools such as the properties of limits, the binomial theorem for series expansions, and the properties of sums. Some sequences will converge to a specific value, while others might diverge. The epsilon-delta definition of a limit is used to find an appropriate n_0 threshold for a given ϵ>

Here's a summary of the limits for the provided sequences:

1. The limit of 1/√n as n approaches infinity is 0.
2. The limit of (2n+1)/(n+1) as n approaches infinity is 2.
3. The limit of (5n-1)/(7n+2) as n approaches infinity is 5/7.
4. The limit of 1/(n-√n) as n approaches infinity is 0.
5. The limit of the sum (1+…+n)/n², which can be expressed as n(n+1)/2n², simplifies to 1/2 as n approaches infinity.
6. The limit of (√1 +√2 +…+√n)/n³ as n approaches infinity is 0.
7. The limit of nċ√(1+(1/n)) - 1 as n approaches infinity is 1/2.
8. The limit of √n² + 1 + √n² - 1 - 2n as n approaches infinity is 0.