Question

Explain why the sequence with terms ( aₙ=sin (nπ/n+1)) is convergent.

Answer 1

The sequence
`\(a_n = \sin\left((n\pi)/(n+1)\right)\)` is convergent.

Step-by-step explanation:

This sequence can be rewritten as
`\(a_n = \sin\left((\pi)/(1+(1)/(n))\right)\)` . As
`\(n\)` approaches infinity,
`\((1)/(n)\)` tends towards zero, making the argument of the sine function approach
`\(\pi\)` . For
`\(n\)` , larger values lead to
`\((1)/(n)\)` being negligibly small, resulting in the argument of the sine function being extremely close to
`\(\pi\)` . The sine of an angle close to
`\(\pi\)` is zero. Therefore, as
`\(n\)` tends towards infinity,
`\(a_n\)` tends towards
`\(\sin(\pi) = 0\)` .

This can be proven by considering the limit as
`\(n\)` approaches infinity for
`\(a_n\)` . Taking the limit of
`\(a_n\)` as
`\(n\)` approaches infinity:

`\[\lim_{{n \to \infty}} \sin\left((n\pi)/(n+1)\right) = \sin(\pi) = 0\]`

Using the fact that the sine function is continuous, we can directly substitute the limiting value of the argument,
`\(\pi\)` , to obtain the result
`\(0\)` . Therefore, the sequence
`\(a_n\)` converges to
`\(0\)` as
`\(n\)`approaches infinity.

Answer 2

Main Answer:

The sequence converges because the sine function, bounded between -1 and 1, approaches a fixed value as n tends to infinity.

Step-by-step explanation:

The given sequence (aₙ = sin(nπ/n+1)) is convergent due to the nature of the sine function and the specific form of the sequence. The sine function is bounded between -1 and 1 for any real input, and in this sequence, the argument of the sine function is nπ/(n+1). As n approaches infinity, the term n/(n+1) approaches 1, and consequently, the argument of the sine function approaches nπ. Since the sine function oscillates periodically with a period of 2π, the sequence becomes periodic with each term converging to a specific value within the interval [-1, 1].

This behavior ensures the convergence of the sequence as n tends to infinity. The boundedness of the sine function and the specific structure of the sequence lead to a convergence pattern that is consistent and predictable, providing a mathematical basis for the convergence of the given sequence. In essence, the convergence is a result of the interplay between the periodicity of the sine function and the characteristics of the sequence's form.