Here are the definitions of the terms as used in analysis:

1. Bounded Sequence:

A sequence $(a_n)$ is bounded if there exists a real number $M > 0$ such that for all $n \in \mathbb{N}$, the absolute value of the terms of the sequence satisfies: $|a_n| \leq M.$ In other words, a sequence is bounded if there is some upper and lower bound on the values of its terms, meaning the terms do not go off to infinity (or negative infinity).

2. Convergence of a Sequence:

A sequence $(a_n)$ converges to a real number $L$ (denoted $\lim_{n \to \infty} a_n = L$) if, for every $\epsilon > 0$, there exists a positive integer $N$ such that for all $n \geq N$, the terms of the sequence satisfy: $|a_n - L| < \epsilon.$ This means that the terms of the sequence get arbitrarily close to the value $L$ as $n$ increases, and the sequence "settles" at that value as $n \to \infty$.

3. Convergence of a Series:

A series is the sum of the terms of a sequence. A series is said to converge if the sequence of its partial sums converges to a finite limit. More formally, consider the series: $S = \sum_{n=1}^{\infty} a_n.$ The $n$-th partial sum $S_N$ is defined as: $S_N = \sum_{n=1}^{N} a_n.$ The series $\sum_{n=1}^{\infty} a_n$ converges to a sum $S$ if the sequence of partial sums $(S_N)$ converges to $S$, i.e., if: $\lim_{N \to \infty} S_N = S.$ If the partial sums do not converge, the series is said to diverge.

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Would you like further details or have any specific questions on these concepts?

Here are 5 related questions for further exploration:

1. How can we determine whether a sequence is bounded using the Bolzano-Weierstrass Theorem?
2. What are some common tests for determining the convergence of a series, such as the Ratio Test or Integral Test?
3. How do Cauchy sequences relate to the concept of convergence in analysis?
4. What is the difference between pointwise convergence and uniform convergence of a sequence of functions?
5. How does the concept of absolute convergence differ from conditional convergence in series?

Tip: When working with sequences and series, it's essential to be familiar with various convergence tests (e.g., the comparison test, the ratio test) to analyze the behavior of series effectively.