Hello ,i shar withe you nice problem show that ∀k∈N^∗ ∃n∈N such that k≤Σ


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Question Number 73378 by mind is power last updated on 10/Nov/19

$H e l l o, i s h a r w i t h e y o u n i c e p r o b l e m$ $s h o w t h a t \forall k \in N^{*} \exists n \in N s u c h t h a t$ $k ⩽ \underset{j = 1}{\sum^{n}} \frac{1}{j} < k + 1$ $h a v e a v e r y N i c e d a y$
Commented byMJS last updated on 11/Nov/19
$H_n =Σ_(j=1) ^n (1/j); lim_(n→+∞) H_n =+∞ ∀n∈N: { ((H_(n+1) −H_n <1)),((H_(n+2) −H_(n+1) <H_(n+1) −H_n )) :} k=1 1≤H_1 <H_2 <H_3 <2 ⇒ ∀k∈N^★ ∃n∈N:k≤Σ_(j=1) ^n (1/j)<k+1$
$H_{n} = \underset{j = 1}{\sum^{n}} \frac{1}{j}; \lim_{n \to + \infty} H_{n} = + \infty$ $\forall n \in N : {\begin{cases} H_{n + 1} - H_{n} < 1 \\ H_{n + 2} - H_{n + 1} < H_{n + 1} - H_{n} \end{cases}$ $k = 1$ $1 ⩽ H_{1} < H_{2} < H_{3} < 2$ $\Rightarrow \forall k \in N^{★} \exists n \in N : k ⩽ \underset{j = 1}{\sum^{n}} \frac{1}{j} < k + 1$
Commented bymind is power last updated on 11/Nov/19

$n i c e s o l u t i o n$
Commented byMJS last updated on 11/Nov/19

$thank you$
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