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)} = e^15/2
because
lim (1 + 1/n)ⁿ = e

Example 2. Find the limit of a sequence:

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

when n tends to +∞

sequence limit:

lim n[(n+1)^1/2 - n^1/2][(n+1)^1/2 + n^1/2]/[(n+1)([n+1]^1/2 + n^1/2)] =
lim n/[(n+1)([n+1]^1/2 + n^1/2)] = lim n/[n+1] 1/[n^1/2[(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:

n^2/3/[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:

n^1/n

when n tends to +∞

sequence limit:

lim e^{1/n ln(n)} = 1

Example 9. Find the limit of a sequence:

[ln(n)]^1/n

when n tends to +∞

sequence limit:

lim e^{1/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 - a^n-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 1^1/n 2^1/n*...*n^1/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/n^1/2)/[1/n^1/2] = 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 =
n^1/2(n - (n² - 1)^1/2)^1/2(n + (n² - 1)^1/2)^1/2/(n + (n² - 1)^1/2)^1/2 =
n^1/2(n² - (n² - 1)^1/2)^1/2/[n^1/2(1 + (1 - 1/n²)^1/2)^1/2] =
1/(1 + (1 - 1/n²)^1/2)^1/2 = 1/2^1/2

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:

cos^3k(2kπ) = 1

sub - sequence limit for n = 3k+1:

cos^3k+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.