# Which of the following assertions is true for the given

Which of the following assertions is true for the given sequences?
1)$\text{Double exponent: use braces to clarify}$ converges
2)$\left\{\frac{n+1}{2n+3}\right\}$ diverges
3)$\left\{-1+2{\left(-1\right)}^{n}\right\}$ diverges
4)$\left\{n\mathrm{sin}\frac{1}{n}\right\}$ converges
5)None of these
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Rylee Marshall

True sequences:
2) $\frac{n+1}{2n+3}$ diverges
3) $-1+2{\left(-1\right)}^{n}$ diverges
${a}_{n}=\frac{n+1}{2n+3}$
Now, check convergence of $\sum _{n=0}^{\mathrm{\infty }}{a}_{n}$
$\sum _{n=0}^{\mathrm{\infty }}\frac{n+1}{2n+3}$
Apply series Divergence test
=diverges
Thus, $\sum _{n=0}^{\mathrm{\infty }}\frac{n+1}{2n+3}:$ diverges

This is a true sequence
3)${a}_{n}=-1+2{\left(-1\right)}^{n}$
Check the convergence of $\sum _{n=0}^{\mathrm{\infty }}{a}_{n}$
$\sum _{n=0}^{\mathrm{\infty }}-1+2{\left(-1\right)}^{n}$
Apply Divergence test
=diverges
Thus, $\sum _{n=0}^{\mathrm{\infty }}a-1+2{\left(-1\right)}^{n}:÷er\ge s$. Thus, this is a true sequence.