Substitute n=1, 2, 3, 4, 5 and find the first five sequences in sequence \(\left\{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dotsm+\frac{1}{2^{n}}\right\}\)

Do the first 6 terms of the sequence \(\displaystyle{g}_{{1}}={2}\) \(\text{and} \ g_2=1\). The rest of the terms are given by the formula \(\displaystyle{g}_{{n}}={n}{g}_{{{n}-{1}}}+{g}_{{{n}-{2}}}\)

Find the limits of the sequences 1)\(\left\{\frac{n^{2}+3}{n^{3}+n^{2}-1}\right\}^{\infty}_{n=1}\) 2)\(\left\{n\sin\frac{\pi}{n}\right\}^{\infty}_{n=1}\)