Question-217895

Question Number 217895 by hardmath last updated on 23/Mar/25

Commented by mr W last updated on 23/Mar/25

$${question}\:{is}\:{wrong}.\:{C}_{{n}} ^{{r}} \:{is}\:{only}\:{defined} \\ $$$${for}\:\mathrm{0}\leqslant{r}\leqslant{n}. \\ $$

Commented by hardmath last updated on 23/Mar/25

$$ \\ $$RHS must be 1/2^(n+1)

Commented by mr W last updated on 23/Mar/25

$${you}\:{didn}'{t}\:{answer}\:{my}\:{question}. \\ $$$${say}\:{n}=\mathrm{10},\:{what}\:{is}\:{meant}\:{with}\:{C}_{\mathrm{10}} ^{\mathrm{100}} ? \\ $$

Commented by Frix last updated on 23/Mar/25

$$\mathrm{I}\:\mathrm{think}\:\mathrm{some}\:\mathrm{people}\:\mathrm{use}\:{C}_{{k}} ^{{n}} \:\mathrm{but}\:\mathrm{others}\:{C}_{{n}} ^{{k}} … \\ $$

Commented by mr W last updated on 23/Mar/25

$${for}\:{k}=\mathrm{1}\:{to}\:\infty,\:{no}\:{matter}\:{whether}\: \\ $$$${you}\:{use}\:{C}_{{n}} ^{{k}} \:{or}\:{you}\:{use}\:{C}_{{k}} ^{{n}} ,\:{it}'{s}\:{the} \\ $$$${same}\:{problem}.\:{if}\:{your}\:{question}\:{is} \\ $$$${not}\:{clear},\:{you}\:{can}\:{not}\:{expect}\:{a}\:{right} \\ $$$${solution}. \\ $$

Commented by hardmath last updated on 23/Mar/25

$$ \\ $$Using (-1)ᵏ = i²ᵏ and (-1)ᵏ⁺¹ = i²ᵏ⁺²:

[∑ₖ[1→+∞] (-1)ᵏ C(2k, n)]²
+ [∑ₖ[1→+∞] (-1)ᵏ⁺¹ C(2k-1, n)]²
= [∑ₖ[1→+∞] i²ᵏ C(2k, n)]²
+ i³.[∑ₖ[1→+∞] i²ᵏ⁻¹ C(2k-1, n)]²
= [Re{(1+i)ⁿ}]² + i⁴ [Im{(1+i)ⁿ}]²
= |(1+i)ⁿ|²
= |1+i|²ⁿ
= (√2)²ⁿ
= 2ⁿ

Answered by SdC355 last updated on 23/Mar/25

$$\mathrm{We}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}\:\mathrm{what}\:\mathrm{sequence}\:\left\{{C}_{{n}} ^{{k}} \right\}\:\mathrm{is}….. \\ $$

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