Solving Sequence and Series Limits: xn = 1/ln(n+1)

[hsong9]

Homework Statement

Let x_n = 1/ln(n+1) for n in N.

a) Use the definition of limit to show that lim(x_n) = 0.
b) Find a specific value of K(ε) as required in the definition of limit for each of i)ε=1/2, and ii)ε=1/10.

The Attempt at a Solution

a) If ε > 0 is given,
1/ln(n+1) < ε <=> ln(n+1) > 1/ε <=> e^(ln(n+1)) > e^(1/ε) <=> n+1 > e^(1/ε)
<=> n > e^(1/ε) - 1
Because ε is arbitrary number, so we have n > 1/ε.
If we choose K to be a number such that K > 1/ε, then we have 1/ln(n+1) < ε for any n > K.

right??

b) so.. K can be 3 for )ε=1/2, and 11 for ii)ε=1/10.
Correct?

 

[Mark44]

Homework Statement

Let x_n = 1/ln(n+1) for n in N.

a) Use the definition of limit to show that lim(x_n) = 0.
b) Find a specific value of K(ε) as required in the definition of limit for each of i)ε=1/2, and ii)ε=1/10.

The Attempt at a Solution

a) If ε > 0 is given,
1/ln(n+1) < ε <=> ln(n+1) > 1/ε <=> e^(ln(n+1)) > e^(1/ε) <=> n+1 > e^(1/ε)
<=> n > e^(1/ε) - 1
Because ε is arbitrary number, so we have n > 1/ε.

You did fine up to the line above. You want n > e^(1/ε). That affects your answers to part b.

If we choose K to be a number such that K > 1/ε, then we have 1/ln(n+1) < ε for any n > K.

right??

b) so.. K can be 3 for )ε=1/2, and 11 for ii)ε=1/10.
Correct?