MAT102 Lecture 19B

• We want to prove $P(n)$ for all $n\in {\mathbb{Z}}^{+}$
• Let $n$ be arbitrary, then prove $P(n)$. If you can't do this, then do Math Induction
• Math Induction
  • Base Case:
    • $n=1$
  • Induction Hypothesis:
    • Assume $P(k)$ is true
  • Induction Step:
    • Prove $P(k)⟹P(k+1)$
• Strong Math Induction
• Structural Math Induction
• 1
  • we have $\sum _{i=1}^{15}\frac{2\cdot (-1{)}^i}{(i+1{)}^2}$
  • Reindex to $0$
  • $k=i-1$
    • $\sum _{k=0}^{14}\frac{2(-1{)}^k}{(k+1{)}^2}$
    • $\sum _{k=0}^{14}\frac{2(-1{)}^{k+1}}{(k+1+1{)}^2}$
    • $\sum _{k=0}^{14}\frac{2(-1)(-1{)}^k}{(k+2{)}^2}$
    • $\sum _{k=0}^{14}\frac{-2(-1{)}^k}{(k+2{)}^2}$
  • $\ell =i+1$
    • $\ell -1=i$
    • $\sum _{l=2}^{16}\frac{2(-1{)}^{\ell -1}}{(l-1+1{)}^2}$
    • $\sum _{l=2}^{16}\frac{-2(-1{)}^{\ell }}{l^2}$
    • #tk try re-indexing from $15\to 1$
• 2
  • Showing that $\sum _{i=1}^{2^n}\frac{1}{i}\ge 1+\frac{n}{2}$ for $n>0$
  • #tk

• 4