The limit of a sequence $\left\{ {{a}_{n}} \right\}$ as $n$ approaches infinity is the valu

Question

The limit of a sequence $\left\{ {{a}_{n}} \right\}$ as $n$ approaches infinity is the value that the terms of the sequence get arbitrarily close to $L$ as $n$ becomes sufficiently large. $\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=L$ $\qquad$ or $\qquad$ ${{a}_{n}}\to L$ as $n\to \infty $ If such an $L$ exists, the sequence is said to converge to $L$ (Convergent sequence). If no such $L$ exists, the sequence diverges (Divergent sequence). When describing the behaviour of a sequence, it's important to determine whether the sequence is convergent or divergent In addition to convergence and divergence, we should consider other prominent behaviours of the sequence, such as whether it is increasing, decreasing, alternating, or constant. Increasing Sequence : Each term is greater than or equal to the preceding term. Decreasing Sequence : Each term is less than or equal to the preceding term. Constant Sequence : All terms are the same. E.g: $3$, $3$, $3$, $3$, …. Alternating Sequence : The terms alternate in sign or exhibit another form of regular alternation. E.g: $-1$, $1$, $-1$, $1$, … Note: An alternating sequence is a sequence where the signs of the terms alternate between positive and negative. This can be achieved by incorporating a factor of ${{\left( -1 \right)}^{n}}$ in the recurrence relation. E.g: ${{u}_{n+1}}={{\left( -1 \right)}^{n}}{{u}_{n}}$

Accepted Answer

The sequence converges to $L=8.1$.

Behaviour	Description
Increasing sequence	Each term is greater than or equal to the preceding term.
Decreasing sequence	Each term is less than or equal to the preceding term.
Constant sequence	All terms are the same, for example \(3,3,3,3,\ldots\).
Alternating sequence	The terms alternate in sign or exhibit another regular alternation, for example \(-1,1,-1,1,\ldots\).

Limits and Behaviour of Sequences

Limits and Behaviour of Sequences

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