Real Sequences

📋 Course Outline

1. Arithmetic Sequences
2. Geometric Sequences
3. Bounds and Monotonicity
4. Convergence and Composition
5. Limits and Adjacent Sequences

📖 1. Arithmetic Sequences

🔑 Key Concepts & Definitions

• Arithmetic sequence: a sequence with a constant reason r such that, for every natural number n, $u_{n+1}-u_n=r$

★ Must-know

📐 Formula — For an arithmetic sequence and natural numbers n and m, $u_n=u_m+(n-m)r$, and in particular $u_n=u_0+nr=u_1+(n-1)r$.

Further detail

📐 Formula — The sum of consecutive terms of an arithmetic sequence is $\sum_{k=p}^{n}u_k=\frac{(n-p+1)(u_p+u_n)}{2}$.

💡 Memory Hook

Add the same reason r at every step: uₙ₊₁ = uₙ + r.

📖 2. Geometric Sequences

🔑 Key Concepts & Definitions

• Geometric sequence: a sequence with a constant reason q such that, for every natural number n, $u_{n+1}=q\,u_n$

★ Must-know

📐 Formula — For a geometric sequence and natural numbers n and m, $u_n=q^{n-m}u_m$, and in particular $u_n=q^nu_0=q^{n-1}u_1$.

• The limit of $q^n$ as n tends to infinity is +∞ if q>1, 0 if −1<q<1, and does not exist if q≤−1.

Further detail

📐 Formula — If $q\ne1$, the sum of consecutive terms of a geometric sequence is $\sum_{k=p}^{n}u_k=u_p\frac{1-q^{n-p+1}}{1-q}$.