Collection of Maths Problems

Alternating and constant sequences

Task number: 2865

• Variant 1

  \(\displaystyle \lim_{n\to\infty} (-1)^n\)

• Resolution

  We will show that the limit does not exist. We may choose the subsequence \((-1)^{2k}\), for which \[\lim_{k\to\infty} (-1)^{2k} = \lim_{k\to\infty} 1 = 1.\] Similarly we may choose the subsequence \((-1)^{2k+1}\) and we have \[\lim_{k\to\infty} (-1)^{2k+1} = \lim_{k\to\infty} -1 = -1.\] If the limit existed, these two values would necessarily be the same, but they are different.

• Result

  The limit does not exist.

• Variant 2

  \(\displaystyle \lim_{n\to\infty} \cos(-1)^n\)

• Resolution

  The cosine function is even, so \(\cos(-1)^n = \cos 1\). So we have \[ \lim_{n\to\infty} \cos(-1)^n = \lim_{n\to\infty} \cos 1 = \cos 1. \]

• Result

  \(\cos 1\)

• Variant 3

  \(\displaystyle \lim_{n\to\infty} (-1)^{n!}\)

• Resolution

  For \(n \geq 2\), \(n!\) is an even number. So \((-1)^{n!} = 1\) for \(n \geq 2\). So we have \[ \lim_{n\to\infty} (-1)^{n!} = \lim_{n\to\infty} 1 = 1. \]

• Result

  1

• Variant 4

  \(\displaystyle \lim_{n\to\infty} \frac{(-1)^n}n\)

• Hint

  First calculate the limit \(\displaystyle \lim_{n\to \infty} \frac1n\)

• Resolution

  We first note that \[\lim_{n\to\infty} \frac1n = 0,\] because this is a limit of type \(\frac{1}{\infty}\).

  Now we use the following fact: for any sequence, if \(\displaystyle \lim_{n\to \infty} |a_n| = 0\), then also \(\displaystyle \lim_{n\to \infty} a_n = 0\). This implies that the limit we are seeking is \(0\).

  The fact above is a consequence of the squeeze lemma. That is \[ \lim_{n\to\infty} -|a_n| \leq \lim_{n\to\infty} a_n \leq \lim_{n\to\infty} |a_n| \] and the limits on the left and right are \(0\).

• Variant 5

  \(\displaystyle \lim_{n\to\infty} \cos\left(\frac{n\pi}4\right)\)

• Hint

  Use a similar approach as in variant 1.

• Resolution

  For \(n = 8k\) we have a subsequence \(\{\cos(2\pi k)\}\) with limit \(1\). For \(n = 8k + 4\) we have a subsequence \(\{\cos((2\pi + 1)k\}\) with limit \(-1\). So the limit does not exist.

• Result

  The limit does not exist.