Mathematical induction (inequality for harmonic numbers)

[glass obsidian]

How did this 1/(2^k + 1) + ....+ 1/(2^(k+1))(red circle in the image)become 2^k • 1/(2^(k+1))(yellow circle in the image)?

[queen wave]

this is hard 🤔

y

waths

[old lark]

you can see it this way

the author murdered the 1/(2^k + 1) + ... + terms

and turned them into something less than that

1/2^(k + 1)

which happens like 2^k - 1 times

but there's also 1/2^(k + 1) on the other end

so it'd be 2^k times

this also means the inequality would simplify to what's on the yellow circle

or orange

idk what color that is

[glass obsidian]

What if 2^k is the number of terms between 1/(2^k + 1) and 1/2^(k+1) (red circle)?

[old lark]

well it'd still be 2^k

what i was talking about was this

the terms between 1/(2^k + 1) + ... + 1/(2^(k + 1) - 1)

would be 2^k - 1

but 1/2^(k + 1) is a term

so it'd be 2^k - 1 + 1 = 2^k