General Principle of Convergence

Suppose that (x_n) is a sequence satisfying

|x_{n+1} - x_n| < (1/2)^n

for each n. Show that (x_n) converges.

(Hint: You would be well advised to use the General Principle of Convergence).

Reply 1

Neapolitan

Reply 2

karola

But why is that :
1/2^(n-1) + ... + 1/2^m < 1/2^(m-1)

Reply 3

Kolya

karola

But why is that :
1/2^(n-1) + ... + 1/2^m < 1/2^(m-1)

This is an old thread! Anyway, I'm not sure it was what the other poster meant, but you can sum the series on the LHS as a G.P.

Reply 4

Ewan

karola

But why is that :
1/2^(n-1) + ... + 1/2^m < 1/2^(m-1)

Hes missed out loads of steps, check your notes from fridays lecture on Analysis (I assume by the bump of this thread your from UCL :p:

)

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