very difficult sequence

[logistic_guy]

here is the question

The sequence \(\displaystyle z_n = -1+i\frac{(-1)^n}{n^2}\) converges to...?

my attemb
we learn to solve sequence in three different ways
first \(\displaystyle |z_n - z| < \epsilon\)
second \(\displaystyle \lim_{n\to \infty} z_n\)
third polar coordinate \(\displaystyle r_n = |z_n|\)

all three ways should give the same answer. why polar coordinate give wrong result for this sequence?

\(\displaystyle \lim_{n\to \infty } Arg \ z_n\) don't exist

 

[Steven G]

What answer did you get?
Did you write out the 1st few terms?
In mathematics, you don't go directly to formulas or techniques all the time. Sometimes (preferably all the time) you need to think about a problem as the results may be easy/obvious. Then you prove the results.

 

[logistic_guy]

What answer did you get?

polar give me 1

Did you write out the 1st few terms?

\(\displaystyle z_0 = -1+i\frac{(-1)^0}{0^2}\)
\(\displaystyle z_1 = -1+i\frac{(-1)^1}{1^2}\)
\(\displaystyle z_2 = -1+i\frac{(-1)^2}{2^2}\)
\(\displaystyle z_3 = -1+i\frac{(-1)^3}{3^2}\)
\(\displaystyle z_4 = -1+i\frac{(-1)^4}{4^2}\)
\(\displaystyle z_5 = -1+i\frac{(-1)^5}{5^2}\)
\(\displaystyle z_6 = -1+i\frac{(-1)^6}{6^2}\)
\(\displaystyle z_7 = -1+i\frac{(-1)^7}{7^2}\)
\(\displaystyle z_8 = -1+i\frac{(-1)^8}{8^2}\)
\(\displaystyle z_9 = -1+i\frac{(-1)^9}{9^2}\)
\(\displaystyle z_{10} = -1+i\frac{(-1)^{10}}{10^2}\)

In mathematics, you don't go directly to formulas or techniques all the time. Sometimes (preferably all the time) you need to think about a problem as the results may be easy/obvious. Then you prove the results.

i don't understand this

 

[Steven G]

Now simplify it!!! How else will you see a pattern!!
What do you think the sequence will converge to? Where your work showing that polar gives you 1??????????

 

[logistic_guy]

Now simplify it!!! How else will you see a pattern!!

\(\displaystyle z_0 = -1+i\frac{1}{0}\)
\(\displaystyle z_1 = -1-i\frac{1}{1}\)
\(\displaystyle z_2 = -1+i\frac{1}{4}\)
\(\displaystyle z_3 = -1-i\frac{1}{9}\)
\(\displaystyle z_4 = -1+i\frac{1}{16}\)
\(\displaystyle z_5 = -1-i\frac{1}{25}\)
\(\displaystyle z_6 = -1+i\frac{1}{36}\)
\(\displaystyle z_7 = -1-i\frac{1}{49}\)
\(\displaystyle z_8 = -1+i\frac{1}{64}\)
\(\displaystyle z_9 = -1-i\frac{1}{81}\)
\(\displaystyle z_{10} = -1+i\frac{1}{100}\)

What do you think the sequence will converge to?

-1

Where your work showing that polar gives you 1??????????

\(\displaystyle \lim_{n\to \infty} r_n = \lim_{n\to \infty} \sqrt{1 + \frac{1}{n^4}} = \sqrt{1 + 0} = \sqrt{1} = 1\)