# Courses in Math. Sequence. Limit. Analytic Examples.

This page helps you learn how to find the limit of a sequence in the analytic way.

## Sequence Limit - Analytic Examples.

### Example 1. Find the limit of a sequence:

[(2n + 1)/(2n - 2)](5n-1)

when n tends to +∞

sequence limit:
lim [1 + 3/(2n-2)](5n-1) = lim [1 + 1/[(2n - 2)/3]](2n - 2)/(2n - 2)(5n - 1) =
lim [[1 + 1/[(2n - 2)/3]](2n - 2)/3]3(5n - 1)/(2n - 2) = e15/2
because
lim (1 + 1/n)n = e

### Example 2. Find the limit of a sequence:

n[(n + 1)1/2 - n1/2]/(n + 1)

when n tends to +∞

sequence limit:
lim n[(n+1)1/2 - n1/2][(n+1)1/2 + n1/2]/[(n+1)([n+1]1/2 + n1/2)] =
lim n/[(n+1)([n+1]1/2 + n1/2)] = lim n/[n+1] 1/[n1/2[(1 + 1/n)1/2 + 1]] = 0

### Example 3. Find the limit of a sequence:

(n!)2/n2n

when n tends to +∞

sequence limit:
lim [(n!)/nn]2 = 0
because of: 1*2*3*…*n < n*n*n*...*n

### Example 4. Find the limit of a sequence:

[(2n2 - n)/(2n2 + 1)]n2

when n tends to +∞

sequence limit:
lim [1 + (-n - 1)/(2n2 + 1)]n2 = lim [1 + 1/[(2n2 + 1)/(-n - 1)]]n2 =
lim [1 + 1/[(2n2 + 1)/(-n - 1)]][(2n2 + 1)/(-n - 1)] [(-n - 1)/(2n2 + 1)]n2 =
e-∞ = 0

### Example 5. Find the limit of a sequence:

[(n+1)/(n-2)]3n-1

when n tends to +∞

sequence limit:
lim [1 + 3/(n - 2)](3n - 1) = lim [1 + 1/[(n - 2)/3]](3n - 1) =
lim [1 + 1/[(n - 2)/3]](n - 2)/3 (3n - 1) 3/(n - 2) = e9

### Example 6. Find the limit of a sequence:

n2/3/[n+1]

when n tends to +∞

sequence limit:
lim [n2/n3]1/3/[1 + 1/n] = 0

### Example 7. Find the limit of a sequence:

ln(n)/n

when n tends to +∞

sequence limit (Stolz theorem):
lim [ln(n) - ln(n - 1)] = 0

### Example 8. Find the limit of a sequence:

n1/n

when n tends to +∞

sequence limit:
lim e1/n ln(n) = 1

### Example 9. Find the limit of a sequence:

[ln(n)]1/n

when n tends to +∞

sequence limit:
lim e1/n ln(ln(n)) = 1

### Example 10. Find the limit of a sequence:

an/n

when n tends to +∞ and a > 1

sequence limit (Stolz theorem):
lim an - an-1 = lim an(1 - 1/a) = +∞

### Example 11. Find the limit of a sequence:

e(1 - 1/(n+1))n

when n tends to +∞

sequence limit:
lim e(1 + 1/(-n-1))(-n-1)n/(-n-1) = e e-1 = 1

### Example 12. Find the limit of a sequence:

[n! en/nn]1/n

when n tends to +∞

sequence limit:
lim e/n (n!)1/n = 0
because
lim (1*2*...*n)1/n = lim 11/n 21/n*...*n1/n = 1

### Example 13. Find the limit a sequence:

when n tends to +∞

lim sin(n)/n = 0

### Example 14. Find the limit

when n tends to +∞

lim arctg(1/n1/2)/[1/n1/2] = 1

### Example 15. Find the limit of a sequence:

[n(n - (n2 - 1)1/2)]1/2

when n tends to +∞

sequence limit:
[n(n - (n2 - 1)1/2)]1/2 = [n(n - (n2 - 1)1/2)]1/2 =
n1/2(n - (n2 - 1)1/2)1/2(n + (n2 - 1)1/2)1/2/(n + (n2 - 1)1/2)1/2 =
n1/2(n2 - (n2 - 1)1/2)1/2/[n1/2(1 + (1 - 1/n2)1/2)1/2] =
1/(1 + (1 - 1/n2)1/2)1/2 = 1/21/2

### Example 16. Find the limit a sequence:

when n tends to +∞

(-1)(n + 1) → there is no limit, however see cases below

## Sub-sequences – limit.

### Example 1. Find the limit of a sequence:

(-1)n-1(2 + 3/n)

when n tends to +∞. Let us consider cases:

sub - sequence limit for n = 2k (even numbers):
-(2 + 3/(2k)) = -2
sub - sequence limit for n = 2k+1 (odd numbers):
(2 + 3/(2k+1)) = 2

### Example 2. Find the limit of a sequence:

1 + n/(n + 1) cos(nπ/2)

when n tends to +∞

sub - sequence limit for n = 2k (even numbers):
1 + 2k/(2k + 1) cos(2kπ/2) = 1 + 2k/(2k + 1) cos(kπ) = 1 + 1 = 2
sub - sequence limit for n = 2k+1 (odd numbers):
1 + (2k + 1)/(2k + 2) cos((2k+1)π/2) = 1 + (2k + 1)/(2k + 2) cos((k + 1/2)π) = 1 + 0 = 1

### Example 3. Find the limit of a sequence:

(n - 1)/(n + 1) cos(2nπ/3)

when n tends to +∞

sub - sequence limit for n = 3k:
(3k - 1)/(3k + 1) cos(2kπ) = 1
sub - sequence limit for n = 3k + 1:
3k/(3k + 2) cos((2k + 2/3)π) = -1/2

### Example 4. Find the limit of a sequence:

(-1)nn

when n tends to +∞

sub - sequence limit for n - even numbers:
(-1)nn = +∞
sub - sequence limit for n - odd numbers:
(-1)nn = -∞

### Example 5. Find the limit of a sequence:

n[(-1)nn]

when n tends to +∞

sub - sequence limit for n - even numbers:
n[(-1)nn] = + ∞
sub - sequence limit for n - odd number:
n[(-1)nn] = 0

### Example 8. Find the limit of a sequence:

1 + n sin(nπ/2)

when n tends to +∞

sub - sequence limit for n = 2k:
1 + 2k sin(kπ) = 1
sub - sequence limit for n = 2k + 1 (k - even number):
1 + (2k + 1) sin((k + 1/2)π) = + ∞
sub - sequence limit for n = 2k + 1 (k - odd number):
1 + (2k + 1) sin((k + 1/2)π) = - ∞

### Example 7. Find the limit of a sequence:

n2/[1 + n2] cos(2nπ/3)

when n tends to +∞

sub - sequence limit for n = 3k:
9k2/[1 + 9k2] cos(2kπ) = 1
sub - sequence limit for n = 3k + 1 (k - even number):
[9k2 + 6k + 1]/[9k2 + 6k + 2] cos(2kπ + 2π/3) = 1/2
sub - sequence limit for n = 3k + 1 (k - odd number):
[9k2 + 6k + 1]/[9k2 + 6k + 2] cos(2kπ + 2π/3) = -1/2

### Example 8. Find the limit of a sequence:

[1 + 2[n(-1)n]]1/n

when n tends to +∞

sub - sequence limit for n - even numbers:
[1 + 2n]1/n = 2
sub - sequence limit for n - odd numbers:
[1 + 2-n]1/n = 1

### Example 9. Find the limit of a sequence:

cosn(2nπ/3)

when n tends to +∞

sub - sequence limit for n = 3k:
cos3k(2kπ) = 1
sub - sequence limit for n = 3k+1:
cos3k+1((k + 1/3)2π) = 0

### Function. Limit.

If you want to find more analytic examples how to calculate limits go to the page function - limits - analytic examples.

### Series. Convergence.

To train another analytic examples containing limits calculations go to the page series - convergence - analytic examples.